Smith Normal Forms and Matrix Theory over Ordered Pair of Normalized Real Numbers

1. Introduction

The theory of Ordered Pair of Normalized Real numbers (OPNs) was introduced by Zhou as a closed arithmetic framework for ordered pairs whose two entries lie in $(0,1)$ [1]. The original construction gives arithmetic operations, inverse operations, powers, roots, elementary functions, ordering, and Cauchy–Schwarz type inequalities in normalized coordinates. The present paper keeps the same normalized setting but reorganizes the algebraic mechanism behind it. The original formulas are closed in ${(0,1)}^2$, but their coordinate expressions, especially multiplication and inverse formulas, are not transparent until the underlying spectral coordinates are separated.

The algebraic core of these formulas is the spectral decomposition induced by $S_{\phi }$. This decomposition leads to a matrix theory over ${\mathcal{O}}_{\phi }$, including matrix equivalence, Smith normal forms, determinant theory, linear systems, similarity, and symmetric congruence. The matrix-theoretic background is the classical theory of matrix equivalence, canonical forms, determinants, invariant factors, and spectral decompositions over fields and principal ideal domains [2,3,4,5]. The module-theoretic background is the structure theory of modules over principal ideal domains and the associated Smith invariant-factor theory [6,7,8,9,10,11].

The scalar observation is simple but powerful, but the main point of the present version is to push it beyond a formal two-channel translation. The paper identifies the intrinsic split semisimple algebra behind OPNs, proves finite module and finite presentation classification theorems, and shows that the four-pivot rank geometry is unavoidable rather than cosmetic. In this sense, the present work should be read as an OPNs reconstruction of several standard parts of linear algebra over split semisimple scalar rings, with the normalization map providing a concrete ordered-pair coordinate model.

Once an admissible normalization function $\phi$ is fixed, every OPNs element

$$\alpha =({\mu }_{\alpha },{\nu }_{\alpha })$$

can be represented by two real spectral coordinates

$$p_{\alpha }=-{\phi }^{-1}\left({\mu }_{\alpha }\right)-{\phi }^{-1}\left({\nu }_{\alpha }\right),q_{\alpha }={\phi }^{-1}\left({\mu }_{\alpha }\right)-{\phi }^{-1}\left({\nu }_{\alpha }\right).$$

In these coordinates, scalar multiplication, addition, and multiplication reduce to componentwise operations on $\mathbb{R}\times \mathbb{R}$. The scalar OPNs algebra is therefore a normalized coordinate realization of a two-channel real algebra. This fact explains the existence of two primitive idempotents, two zero-divisor axes, and a complete spectral functional calculus.

The scalar two-channel representation gives the idempotent decomposition. At the matrix level this decomposition produces additional reduction data. A matrix over OPNs has two real spectral components,

$$A=A_p{e}_p+A_q{e}_q.$$

Under row-column equivalence, this decomposition yields a four-pivot reduction theory. A real matrix has only two pivot types, 1 and 0. An OPNs matrix has four:

$$\tilde{1},e_p,e_q,\tilde{0}.$$

The idempotents $e_p$ and $e_q$ represent one-channel pivots. A pivot equal to $11$ constrains both spectral channels, a pivot equal to $e_p$ constrains only the p-channel, a pivot equal to $e_q$ constrains only the q-channel, and a zero pivot constrains neither channel.

The polynomial theory gives the strongest form of the reduction theorem. It is modeled on the classical Smith normal form and the PID module classification theorem [6,7,8,10]. Although ${\mathcal{O}}_{\phi }\left[t\right]$ has zero divisors and is not a principal ideal domain, the primitive idempotents $e_p$ and $e_q$ split ${\mathcal{O}}_{\phi }\left[t\right]$ into two polynomial PID channels. Consequently polynomial matrices over ${\mathcal{O}}_{\phi }\left[t\right]$ admit idempotent Smith normal forms whose invariant factors are OPNs polynomials of the form $f_i\left(t\right)e_p+g_i\left(t\right)e_q$. The four-pivot normal form for ordinary OPNs matrices is the constant-matrix shadow of this polynomial Smith theory. It gives a complete equivalence classification and produces canonical formulas for kernels, images, cokernels, rank defects, determinantal ideals, and solvability of linear systems.

The rest of the paper is organized around a small number of structural reductions. Section 2 develops the scalar spectral algebra, including admissible normalizations, primitive idempotents, zero divisors, and the elementary function calculus. Section 3 proves the matrix splitting principle and the idempotent Smith normal form for polynomial matrices. Section 4 develops the four-pivot rank-pair geometry, including equivalence, stable sums, factorization order, orbit closure, and determinantal recovery. Section 5 treats finite OPNs modules, exact splitting, kernels, images, cokernels, determinantal ideals, and linear systems. Section 6 develops determinant, adjugate, and inverse theory. Section 7 gives characteristic polynomials, invariant factors, rational canonical forms, regular spectral rectangles, and similarity classification. Section 8 treats symmetric matrices, congruence, inertia, and positivity. Section 9 records matrix functions, matrix equations, and gauge equivalence of normalizations. Section 10 concludes.

2. Scalar Spectral Algebra and Elementary Functions

This section fixes the algebraic meaning of the OPNs operations. The normalized pair $(\mu ,\nu )$ is not used as a coordinate system in which multiplication is componentwise. Instead, the normalized pair is first sent by $S_{\phi }$ to the spectral pair $(p,q)$, the direct product algebra structure of $\mathbb{R}\times \mathbb{R}$ is used there, and the result is transported back to ${(0,1)}^2$. This distinction is essential: the idempotents, the zero-divisor axes, and the later four-pivot matrix form are consequences of the transported product algebra, not of ordinary coordinatewise multiplication in normalized coordinates.

Definition 2.1

(Admissible normalization and spectral coordinates). A function

$$\phi :\mathbb{R}⟶(0,1)$$

is called admissible if it is continuous, strictly increasing, onto $(0,1)$, and satisfies

$$\phi \left(x\right)+\phi (-x)=1,x\in \mathbb{R}.$$

(2.1)

For a fixed admissible φ, set ${\mathcal{O}}_{\phi }={(0,1)}^2$. For $\alpha =({\mu }_{\alpha },{\nu }_{\alpha })\in {\mathcal{O}}_{\phi }$, define

$$x_{\alpha }={\phi }^{-1}\left({\mu }_{\alpha }\right),y_{\alpha }={\phi }^{-1}\left({\nu }_{\alpha }\right),$$

and

$$p_{\alpha }=-(x_{\alpha }+y_{\alpha }),q_{\alpha }=x_{\alpha }-y_{\alpha }.$$

(2.2)

The spectral map is

$$S_{\phi }:{\mathcal{O}}_{\phi }⟶{\mathbb{R}}^2,S_{\phi }\left(\alpha \right)=(p_{\alpha },q_{\alpha }).$$

The symmetry condition gives $\phi \left(0\right)=1/2$ and

$${\phi }^{-1}(1-t)=-{\phi }^{-1}\left(t\right),0<t<1.$$

(2.3)

Indeed, putting $x=0$ in (2.1) gives $2\phi \left(0\right)=1$. If $t=\phi \left(x\right)$, then $1-t=\phi (-x)$, and applying ${\phi }^{-1}$ proves (2.3). Solving the linear system (2.2) gives

$$S_{\phi }^{-1}(p,q)=\left(\phi \left({\displaystyle \frac{q-p}{2}}\right),\phi \left(-{\displaystyle \frac{p+q}{2}}\right)\right),(p,q)\in {\mathbb{R}}^2.$$

(2.4)

Thus $S_{\phi }$ is a bijection.

The inverse formula also explains why no special choice of logistic, arctangent, or other normalization is needed. The normalization only provides a smooth or topological chart from real spectral data back into the open square ${(0,1)}^2$. Algebraically, all subsequent operations are performed in the spectral variables and are therefore independent of the particular visual shape of $\phi$.

Throughout the paper, $\mathbb{R}\times \mathbb{R}$ is regarded as the direct product real algebra, with

$$\begin{array}{cccccc}\hfill (p_1,q_1)+(p_2,q_2)& =(p_1+p_2,q_1+q_2),\hfill & \hfill c(p,q)& =(cp,cq),\hfill & \hfill (p_1,q_1)(p_2,q_2)& =(p_1{p}_2,q_1{q}_2).\hfill \end{array}$$

Its zero and unit are $(0,0)$ and $(1,1)$. The following theorem collects the scalar consequences of this choice. It is the scalar engine of the paper: after it is proved, every later construction is obtained by applying ordinary real algebra separately in the two spectral channels and then recombining the results with $e_p$ and $e_q$.

Theorem 2.2

(Scalar spectral algebra theorem). There is a unique real algebra structure on ${\mathcal{O}}_{\phi }$ for which $S_{\phi }$ is an isomorphism from ${\mathcal{O}}_{\phi }$ onto the direct product algebra $\mathbb{R}\times \mathbb{R}$. Explicitly, for $\alpha ,\beta \in {\mathcal{O}}_{\phi }$ and $c\in \mathbb{R}$,

$$\begin{array}{cc}\hfill \alpha +\beta & =S_{\phi }^{-1}(p_{\alpha }+p_{\beta },q_{\alpha }+q_{\beta }),\hfill \\ \hfill c\alpha & =S_{\phi }^{-1}(c{p}_{\alpha },c{q}_{\alpha }),\hfill \\ \hfill \alpha \beta & =S_{\phi }^{-1}(p_{\alpha }{p}_{\beta },q_{\alpha }{q}_{\beta }).\hfill \end{array}$$

(2.5)

The transport identities

$$\begin{array}{cc}\hfill S_{\phi }(\alpha +\beta )& =S_{\phi }\left(\alpha \right)+S_{\phi }\left(\beta \right),\hfill \\ \hfill S_{\phi }\left(c\alpha \right)& =c{S}_{\phi }\left(\alpha \right),\hfill \\ \hfill S_{\phi }\left(\alpha \beta \right)& =S_{\phi }\left(\alpha \right)S_{\phi }\left(\beta \right)\hfill \end{array}$$

(2.6)

hold, with the operations on the right computed componentwise in $\mathbb{R}\times \mathbb{R}$. The additive and multiplicative identities are

$$\tilde{0}=S_{\phi }^{-1}(0,0)=\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right),\tilde{1}=S_{\phi }^{-1}(1,1)=\left(\phi \left(0\right),\phi (-1)\right).$$

(2.7)

Define the primitive idempotents by

$$e_p=S_{\phi }^{-1}(1,0),e_q=S_{\phi }^{-1}(0,1).$$

(2.8)

Then

$${e_p}^2=e_p,e_q^2=e_q,e_p{e}_q=\tilde{0},e_p+e_q=\tilde{1}.$$

(2.9)

Every $\alpha \in {\mathcal{O}}_{\phi }$ has the unique decomposition

$$\alpha =p_{\alpha }{e}_p+q_{\alpha }{e}_q.$$

(2.10)

The two coordinate ideals are

$${\mathrm{I}}_p=\{\alpha \in {\mathcal{O}}_{\phi }:p_{\alpha }=0\}=\mathbb{R}{e}_q,{\mathrm{I}}_q=\{\alpha \in {\mathcal{O}}_{\phi }:q_{\alpha }=0\}=\mathbb{R}{e}_p.$$

(2.11)

The ideals of ${\mathcal{O}}_{\phi }$ are precisely $\left\{\tilde{0}\right\}$, ${\mathrm{I}}_p$, ${\mathrm{I}}_q$, and ${\mathcal{O}}_{\phi }$. An element α is a unit precisely when $p_{\alpha }{q}_{\alpha }\ne 0$, and it is a zero divisor precisely when $p_{\alpha }{q}_{\alpha }=0$. In normalized coordinates the two zero-divisor lines are

$${\mu }_{\alpha }+{\nu }_{\alpha }=1,{\mu }_{\alpha }={\nu }_{\alpha }.$$

(2.12)

Proof. 

The bijection $S_{\phi }$ allows the direct product algebra structure on $\mathbb{R}\times \mathbb{R}$ to be transported to ${\mathcal{O}}_{\phi }$. Formula (2.5) is exactly this pullback definition. Applying $S_{\phi }$ to each formula in (2.5) gives the three identities in (2.6); conversely, the transport identities determine the operations because $S_{\phi }$ is injective. Associativity, commutativity, distributivity, compatibility with real scalar multiplication, and the existence of $\tilde{0}$ and $\tilde{1}$ all follow by applying $S_{\phi }$ and using the corresponding componentwise identities in $\mathbb{R}\times \mathbb{R}$. This proves that ${\mathcal{O}}_{\phi }$ is a commutative real algebra with identity and that $S_{\phi }$ is a real-algebra isomorphism.

The formulas for $\tilde{0}$ and $\tilde{1}$ follow from (2.4). For $\tilde{0}$, one obtains $S_{\phi }^{-1}(0,0)=(\phi \left(0\right),\phi \left(0\right))=(1/2,1/2)$. For $\tilde{1}$, one obtains $S_{\phi }^{-1}(1,1)=(\phi \left(0\right),\phi (-1))$.

The idempotent laws are direct consequences of the componentwise product in $\mathbb{R}\times \mathbb{R}$. For example,

$$S_{\phi }\left({e_p}^2\right)=S_{\phi }\left(e_p\right)S_{\phi }\left(e_p\right)=(1,0)(1,0)=(1,0)=S_{\phi }\left(e_p\right).$$

Injectivity of $S_{\phi }$ gives ${e_p}^2=e_p$. The same calculation gives $e_q^2=e_q$, $e_p{e}_q=\tilde{0}$, and $e_p+e_q=\tilde{1}$ from the identities ${(0,1)}^2=(0,1)$, $(1,0)(0,1)=(0,0)$, and $(1,0)+(0,1)=(1,1)$.

For the decomposition, apply the transport identities for addition and scalar multiplication:

$$S_{\phi }(p_{\alpha }{e}_p+q_{\alpha }{e}_q)=p_{\alpha }{S}_{\phi }\left(e_p\right)+q_{\alpha }{S}_{\phi }\left(e_q\right)=p_{\alpha }(1,0)+q_{\alpha }(0,1)=(p_{\alpha },q_{\alpha }).$$

This is $S_{\phi }\left(\alpha \right)$, so injectivity gives (2.10). If also $\alpha =a{e}_p+b{e}_q$ with $a,b\in \mathbb{R}$, then applying $S_{\phi }$ gives $(p_{\alpha },q_{\alpha })=(a,b)$, so $a=p_{\alpha }$ and $b=q_{\alpha }$. Hence the decomposition is unique.

Under the isomorphism $S_{\phi }$, ideals of ${\mathcal{O}}_{\phi }$ correspond to ideals of $\mathbb{R}\times \mathbb{R}$. Since the only ideals of $\mathbb{R}$ are 0 and $\mathbb{R}$, the only ideals of $\mathbb{R}\times \mathbb{R}$ are $0\times 0$, $0\times \mathbb{R}$, $\mathbb{R}\times 0$, and $\mathbb{R}\times \mathbb{R}$. Their inverse images are respectively $\left\{\tilde{0}\right\}$, ${\mathrm{I}}_p$, ${\mathrm{I}}_q$, and ${\mathcal{O}}_{\phi }$. The identities ${\mathrm{I}}_p=\mathbb{R}{e}_q$ and ${\mathrm{I}}_q=\mathbb{R}{e}_p$ follow from $S_{\phi }\left(e_q\right)=(0,1)$ and $S_{\phi }\left(e_p\right)=(1,0)$.

An element $(p,q)\in \mathbb{R}\times \mathbb{R}$ is a unit precisely when both components are nonzero, with inverse $(1/p,1/q)$. Transporting this statement gives

$${\alpha }^{-1}=S_{\phi }^{-1}\left({\displaystyle \frac{1}{p_{\alpha }}},{\displaystyle \frac{1}{q_{\alpha }}}\right),p_{\alpha }{q}_{\alpha }\ne 0.$$

Similarly, an element of the product algebra is a zero divisor precisely when at least one component is zero. Translating $p_{\alpha }=0$ gives $x_{\alpha }+y_{\alpha }=0$, hence ${\nu }_{\alpha }=\phi (-x_{\alpha })=1-{\mu }_{\alpha }$. Translating $q_{\alpha }=0$ gives $x_{\alpha }=y_{\alpha }$, hence ${\mu }_{\alpha }={\nu }_{\alpha }$. This proves (2.12). □

The preceding theorem also has an intrinsic ring-theoretic interpretation. It is useful to state this explicitly, because it explains why the later matrix theory has exactly two spectral channels and exactly four local pivot types. The following result is not an additional assumption; it is the structural content of the transported algebra.

Theorem 2.3

(Intrinsic split semisimple structure and rigidity). The algebra ${\mathcal{O}}_{\phi }$ is a reduced semisimple Artinian commutative real algebra. Its only central idempotents are

$$\tilde{0},e_p,e_q,\tilde{1},$$

(2.13)

and $e_p,e_q$ are the only nonzero primitive central idempotents. Consequently

$${\mathcal{O}}_{\phi }={\mathcal{O}}_{\phi }{e}_p\oplus {\mathcal{O}}_{\phi }{e}_q=\mathbb{R}{e}_p\oplus \mathbb{R}{e}_q$$

(2.14)

as a direct product of two simple real algebras. Every unital real-algebra automorphism of ${\mathcal{O}}_{\phi }$ either fixes $e_p,e_q$ separately or interchanges them, and therefore

$${Aut}_{\mathbb{R}-alg}\left({\mathcal{O}}_{\phi }\right)\cong C_2.$$

(2.15)

Moreover,

$${Der}_{\mathbb{R}}\left({\mathcal{O}}_{\phi }\right)=0.$$

(2.16)

Proof. 

The isomorphism $S_{\phi }:{\mathcal{O}}_{\phi }\to \mathbb{R}\times \mathbb{R}$ identifies ${\mathcal{O}}_{\phi }$ with the direct product of two copies of the field $\mathbb{R}$. Since a field is reduced, Artinian, and simple as an algebra over itself, the product $\mathbb{R}\times \mathbb{R}$ is reduced, semisimple, and Artinian. Transporting these properties through $S_{\phi }$ proves the first assertion.

An idempotent in $\mathbb{R}\times \mathbb{R}$ is a pair $(u,v)$ satisfying $(u^2,v^2)=(u,v)$. Since the only idempotents in $\mathbb{R}$ are 0 and 1, the only idempotents of the product algebra are $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Their inverse images under $S_{\phi }$ are exactly $\tilde{0},e_p,e_q,\tilde{1}$. The two nonzero proper idempotents $(1,0)$ and $(0,1)$ cannot be decomposed into sums of two nonzero orthogonal idempotents, because each is supported on a single field factor. Hence $e_p$ and $e_q$ are primitive. Conversely, $\tilde{1}=e_p+e_q$ is decomposable and $\tilde{0}$ is not nonzero, so no other nonzero primitive central idempotents exist. Commutativity makes all idempotents central.

The decomposition (2.14) follows from $\tilde{1}=e_p+e_q$ and $e_p{e}_q=\tilde{0}$. The two summands are isomorphic to $\mathbb{R}$ through their nonzero spectral coordinates, hence they are simple real algebras. Any unital real-algebra automorphism preserves $\tilde{0}$, $\tilde{1}$, and primitive central idempotents. Since the primitive central idempotents are exactly $e_p$ and $e_q$, an automorphism either fixes both or interchanges them. Conversely, both possibilities are realized in the product algebra by the identity and the transposition of the two real factors. This proves (2.15).

It remains to prove the rigidity statement for derivations. Let $D:{\mathcal{O}}_{\phi }\to {\mathcal{O}}_{\phi }$ be a real derivation. From ${e_p}^2=e_p$ one obtains

$$D\left(e_p\right)=D\left({e_p}^2\right)=D\left(e_p\right)e_p+e_{p}D\left(e_p\right)=2e_{p}D\left(e_p\right).$$

Write $D\left(e_p\right)=a{e}_p+b{e}_q$ with $a,b\in \mathbb{R}$. The last identity becomes

$$a{e}_p+b{e}_q=2a{e}_p,$$

so $a=b=0$ and $D\left(e_p\right)=0$. The same argument applied to $e_q^2=e_q$ gives $D\left(e_q\right)=0$. Since every element of ${\mathcal{O}}_{\phi }$ has the form $p{e}_p+q{e}_q$ and D is $\mathbb{R}$-linear, D vanishes on all of ${\mathcal{O}}_{\phi }$. Hence (2.16) holds. □

The scalar structure theorem says more than that OPNs support addition and multiplication. It says that any real operation whose domain is meaningful in both spectral channels has a canonical OPNs lift. The original elementary formulas for powers, roots, logarithms, exponentials, and trigonometric functions are all instances of this single channelwise principle.

Proposition 2.4

(Channelwise scalar functional calculus). Let $D\subseteq {\mathbb{R}}^m$ and let $F:D\to \mathbb{R}$ be a real function. For ${\alpha }_1,\dots ,{\alpha }_m\in {\mathcal{O}}_{\phi }$, write

$$S_{\phi }\left({\alpha }_j\right)=(p_j,q_j),1\le j\le m.$$

If $(p_1,\dots ,p_m)$ and $(q_1,\dots ,q_m)$ lie in D, define

$$F_{{\mathcal{O}}_{\phi }}({\alpha }_1,\dots ,{\alpha }_m)=S_{\phi }^{-1}\left(F(p_1,\dots ,p_m),F(q_1,\dots ,q_m)\right).$$

(2.17)

For one variable this gives

$$f_{{\mathcal{O}}_{\phi }}\left(\alpha \right)=S_{\phi }^{-1}\left(f\left(p_{\alpha }\right),f\left(q_{\alpha }\right)\right)=\left(\phi \left({\displaystyle \frac{f\left(q_{\alpha }\right)-f\left(p_{\alpha }\right)}{2}}\right),\phi \left(-{\displaystyle \frac{f\left(p_{\alpha }\right)+f\left(q_{\alpha }\right)}{2}}\right)\right).$$

(2.18)

The lift respects composition whenever the corresponding real compositions are defined. If $f\left(t\right)={\sum }_{k=0}^{\infty }{c}_k{t}^k$ converges at $p_{\alpha }$ and $q_{\alpha }$, then

$$S_{\phi }\left({\displaystyle \sum _{k=0}^{\infty }}{c}_k{\alpha }^k\right)=(f\left(p_{\alpha }\right),f\left(q_{\alpha }\right)).$$

(2.19)

Proof. 

The definition (2.17) is meaningful because $S_{\phi }^{-1}$ is defined on all of ${\mathbb{R}}^2$. For composition, suppose $H\left(x\right)=F(G_1\left(x\right),\dots ,G_m\left(x\right))$ is a real composition. Applying $S_{\phi }$ to the OPNs expression

$$F_{{\mathcal{O}}_{\phi }}\left({\left(G_1\right)}_{{\mathcal{O}}_{\phi }}({\beta }_1,\dots ,{\beta }_n),\dots ,{\left(G_m\right)}_{{\mathcal{O}}_{\phi }}({\beta }_1,\dots ,{\beta }_n)\right)$$

gives a p-channel equal to $H(p_{{\beta }_1},\dots ,p_{{\beta }_n})$ and a q-channel equal to $H(q_{{\beta }_1},\dots ,q_{{\beta }_n})$. These are exactly the two channels of $H_{{\mathcal{O}}_{\phi }}({\beta }_1,\dots ,{\beta }_n)$, so injectivity proves the composition law. For the power series, repeated use of (2.6) gives $S_{\phi }\left({\alpha }^k\right)=(p_{\alpha }^k,q_{\alpha }^k)$ for every $k\ge 0$. Real linearity and convergence in the two channels then give (2.19). □

The elementary operations are recovered from Proposition 2.4. For $\alpha ,\beta \in {\mathcal{O}}_{\phi }$ and $c\in \mathbb{R}$,

$$\begin{array}{cccc}\hfill \alpha +\beta & =S_{\phi }^{-1}(p_{\alpha }+p_{\beta },q_{\alpha }+q_{\beta }),\hfill & \hfill \alpha -\beta & =S_{\phi }^{-1}(p_{\alpha }-p_{\beta },q_{\alpha }-q_{\beta }),\hfill \\ \hfill c\alpha & =S_{\phi }^{-1}(c{p}_{\alpha },c{q}_{\alpha }),\hfill & \hfill \alpha \beta & =S_{\phi }^{-1}(p_{\alpha }{p}_{\beta },q_{\alpha }{q}_{\beta }),\hfill \\ \hfill {\alpha }^{-1}& =S_{\phi }^{-1}\left({\displaystyle \frac{1}{p_{\alpha }}},{\displaystyle \frac{1}{q_{\alpha }}}\right),\hfill & \hfill {\displaystyle \frac{\alpha }{\beta }}& =S_{\phi }^{-1}\left({\displaystyle \frac{p_{\alpha }}{p_{\beta }}},{\displaystyle \frac{q_{\alpha }}{q_{\beta }}}\right).\hfill \end{array}$$

(2.20)

The inverse and quotient formulas require the corresponding denominators to be nonzero. For real powers, roots, exponentials, and logarithms,

$$\begin{array}{cccc}\hfill {\alpha }^a& =S_{\phi }^{-1}(p_{\alpha }^a,q_{\alpha }^a),\hfill & \hfill \sqrt[n]{\alpha }& =S_{\phi }^{-1}\left(\sqrt[n]{p_{\alpha }},\sqrt[n]{q_{\alpha }}\right),\hfill \\ \hfill {exp}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(e^{p_{\alpha }},e^{q_{\alpha }}\right),\hfill & \hfill {log}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(log{p}_{\alpha },log{q}_{\alpha }\right).\hfill \end{array}$$

(2.21)

The real power and root formulas are taken on any real branch on which both channel values are defined, and the logarithm formula requires $p_{\alpha }>0$ and $q_{\alpha }>0$.

The trigonometric functions are

$$\begin{array}{cccc}\hfill {sin}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(sin{p}_{\alpha },sin{q}_{\alpha }\right),\hfill & \hfill {cos}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(cos{p}_{\alpha },cos{q}_{\alpha }\right),\hfill \\ \hfill {tan}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(tan{p}_{\alpha },tan{q}_{\alpha }\right),\hfill & \hfill {cot}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(cot{p}_{\alpha },cot{q}_{\alpha }\right),\hfill \\ \hfill {sec}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(sec{p}_{\alpha },sec{q}_{\alpha }\right),\hfill & \hfill {csc}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(csc{p}_{\alpha },csc{q}_{\alpha }\right).\hfill \end{array}$$

(2.22)

The usual nonvanishing denominator conditions are imposed channel by channel. On the real domains of the inverse trigonometric functions,

$$\begin{array}{cccc}\hfill {arcsin}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(arcsin{p}_{\alpha },arcsin{q}_{\alpha }\right),\hfill & \hfill {arccos}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(arccos{p}_{\alpha },arccos{q}_{\alpha }\right),\hfill \\ \hfill {arctan}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(arctan{p}_{\alpha },arctan{q}_{\alpha }\right),\hfill & \hfill {arccot}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(arccot{p}_{\alpha },arccot{q}_{\alpha }\right).\hfill \end{array}$$

(2.23)

The formulas for ${arcsin}_{{\mathcal{O}}_{\phi }}$ and ${arccos}_{{\mathcal{O}}_{\phi }}$ require $|p_{\alpha }|\le 1$ and $|q_{\alpha }|\le 1$. The hyperbolic functions are

$$\begin{array}{cccc}\hfill {sinh}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(sinh{p}_{\alpha },sinh{q}_{\alpha }\right),\hfill & \hfill {cosh}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(cosh{p}_{\alpha },cosh{q}_{\alpha }\right),\hfill \\ \hfill {tanh}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(tanh{p}_{\alpha },tanh{q}_{\alpha }\right),\hfill & \hfill {coth}_{{\mathcal{O}}_{\phi }}\left(\alpha \right)& =S_{\phi }^{-1}\left(coth{p}_{\alpha },coth{q}_{\alpha }\right).\hfill \end{array}$$

(2.24)

The formula for ${coth}_{{\mathcal{O}}_{\phi }}$ requires $p_{\alpha }\ne 0$ and $q_{\alpha }\ne 0$.

3. Matrix Splitting and Polynomial Smith Theory

The scalar decomposition now passes entrywise to matrices. This passage is formally simple but structurally decisive. It converts matrix questions over ${\mathcal{O}}_{\phi }$ into pairs of real matrix questions, while the recombination by $e_p$ and $e_q$ produces new OPNs-level invariants, most notably the rank pair and the four-pivot profile. The point is not to replace OPNs matrices by two unrelated real matrices; rather, the two real components are the spectral coordinates of a single OPNs matrix.

Let $M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ denote the set of $m\times n$ matrices with entries in ${\mathcal{O}}_{\phi }$. For $A=\left({\alpha }_{ij}\right)\in M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ define real matrices

$$A_p=\left(p_{{\alpha }_{ij}}\right),A_q=\left(q_{{\alpha }_{ij}}\right).$$

Equivalently,

$$A=A_p{e}_p+A_q{e}_q.$$

(3.1)

This formula should be read as a spectral expansion of the whole matrix, not merely as a notation for its entries. Matrix multiplication, invertibility, polynomial evaluation, determinants, and linear systems will all respect this expansion because the idempotents annihilate the mixed terms.

Theorem 3.1

(Matrix splitting and transfer principle). The component map

$$A⟼(A_p,A_q)$$

(3.2)

identifies $M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ with $M_{m,n}\left(\mathbb{R}\right)\times M_{m,n}\left(\mathbb{R}\right)$ as a real vector space. For compatible matrices,

$${\left(AB\right)}_p=A_p{B}_p,{\left(AB\right)}_q=A_q{B}_q.$$

(3.3)

Thus $M_n\left({\mathcal{O}}_{\phi }\right)$ is isomorphic as a real algebra to $M_n\left(\mathbb{R}\right)\times M_n\left(\mathbb{R}\right)$. If $X=X_p{e}_p+X_q{e}_q\in M_n\left({\mathcal{O}}_{\phi }\right)$, then X is invertible over ${\mathcal{O}}_{\phi }$ precisely when $X_p$ and $X_q$ are invertible over $\mathbb{R}$, and then

$$X^{-1}=X_p^{-1}{e}_p+X_q^{-1}{e}_q.$$

(3.4)

Under the same component map, transpose, polynomial evaluation, determinants, kernels, images, cokernels, and linear equations are computed channel by channel.

Proof. 

The entrywise decomposition of the scalar algebra gives the vector-space identification immediately. To prove multiplication, write $A=A_p{e}_p+A_q{e}_q$ and $B=B_p{e}_p+B_q{e}_q$. The idempotent laws give

$$\begin{array}{cc}\hfill AB& =(A_p{e}_p+A_q{e}_q)(B_p{e}_p+B_q{e}_q)\hfill \\ \hfill & =A_p{B}_p{e_p}^2+A_p{B}_q{e}_p{e}_q+A_q{B}_p{e}_q{e}_p+A_q{B}_q{e}_q^2\hfill \\ \hfill & =A_p{B}_p{e}_p+A_q{B}_q{e}_q.\hfill \end{array}$$

This proves (3.3). Addition, scalar multiplication, and transpose split entrywise, so the square case gives the stated algebra isomorphism.

If $X_p$ and $X_q$ are invertible, then (3.3) gives

$$(X_p{e}_p+X_q{e}_q)(X_p^{-1}{e}_p+X_q^{-1}{e}_q)=I{e}_p+I{e}_q=I,$$

and the same computation in the other order gives a two-sided inverse. Conversely, if $Y=Y_p{e}_p+Y_q{e}_q$ is an inverse of X, then $XY=YX=I$ implies

$$X_p{Y}_p=Y_p{X}_p=I,X_q{Y}_q=Y_q{X}_q=I.$$

Thus both real components are invertible, and (28) follows.

For a polynomial $h\left(t\right)=h_p\left(t\right)e_p+h_q\left(t\right)e_q$ and a square matrix $A=A_p{e}_p+A_q{e}_q$, repeated use of (3.3) gives

$$h\left(A\right)=h_p\left(A_p\right)e_p+h_q\left(A_q\right)e_q.$$

The determinant statement follows by applying the Leibniz formula entrywise. Finally, a vector $x=x_p{e}_p+x_q{e}_q$ satisfies $Ax=b$ precisely when $A_p{x}_p=b_p$ and $A_q{x}_q=b_q$. Hence kernels, images, cokernels, and linear equations split into the two real channels. □

The four-pivot normal form for ordinary OPNs matrices is the degree-zero case of a polynomial theory over ${\mathcal{O}}_{\phi }\left[t\right]$. This is the point where the zero divisors of ${\mathcal{O}}_{\phi }$ could have obstructed a Smith theory. The obstruction is avoided not because ${\mathcal{O}}_{\phi }\left[t\right]$ is a PID in the usual sense, but because the idempotents split it into two honest polynomial PID channels. Thus divisibility is not a single-chain phenomenon inside an integral domain; it is a paired divisibility relation in two real polynomial rings.

Theorem 3.2

(Polynomial splitting over OPNs). Every polynomial $h\left(t\right)\in {\mathcal{O}}_{\phi }\left[t\right]$ has a unique representation

$$h\left(t\right)=h_p\left(t\right)e_p+h_q\left(t\right)e_q,h_p\left(t\right),h_q\left(t\right)\in \mathbb{R}\left[t\right].$$

(3.5)

The map

$${\mathcal{O}}_{\phi }\left[t\right]⟶\mathbb{R}\left[t\right]\times \mathbb{R}\left[t\right],h\left(t\right)⟼(h_p\left(t\right),h_q\left(t\right))$$

(3.6)

is an isomorphism of commutative real algebras. If

$$a\left(t\right)=a_p\left(t\right)e_p+a_q\left(t\right)e_q,b\left(t\right)=b_p\left(t\right)e_p+b_q\left(t\right)e_q,$$

then $a\left(t\right)$ divides $b\left(t\right)$ in ${\mathcal{O}}_{\phi }\left[t\right]$ precisely when $a_p\left(t\right)$ divides $b_p\left(t\right)$ and $a_q\left(t\right)$ divides $b_q\left(t\right)$ in $\mathbb{R}\left[t\right]$, with the convention that 0 divides only 0. The units of ${\mathcal{O}}_{\phi }\left[t\right]$ are precisely the elements

$$c_p{e}_p+c_q{e}_q,c_p,c_q\in \mathbb{R}\setminus \left\{0\right\}.$$

(3.7)

A square polynomial matrix $U\left(t\right)=U_p\left(t\right)e_p+U_q\left(t\right)e_q$ is unimodular over ${\mathcal{O}}_{\phi }\left[t\right]$ precisely when $U_p\left(t\right)$ and $U_q\left(t\right)$ are unimodular over $\mathbb{R}\left[t\right]$, and then

$$U{\left(t\right)}^{-1}=U_p{\left(t\right)}^{-1}{e}_p+U_q{\left(t\right)}^{-1}{e}_q.$$

(3.8)

Proof. 

Write every coefficient of $h\left(t\right)={\sum }_i{\alpha }_i{t}^i$ in the form ${\alpha }_i=a_i{e}_p+b_i{e}_q$. Then

$$h\left(t\right)=\left(\sum _i{a}_i{t}^i\right)e_p+\left(\sum _i{b}_i{t}^i\right)e_q,$$

which proves existence of (3.5). If $h_p\left(t\right)e_p+h_q\left(t\right)e_q=0$, then comparison under the scalar decomposition forces all coefficients of $h_p$ and $h_q$ to vanish; hence the representation is unique. Addition and real scalar multiplication are componentwise. Multiplication follows from

$$(h_p{e}_p+h_q{e}_q)(k_p{e}_p+k_q{e}_q)=h_p{k}_p{e}_p+h_q{k}_q{e}_q,$$

because the mixed products contain $e_p{e}_q$ and vanish. Thus (3.6) is an algebra isomorphism.

A polynomial $h=h_p{e}_p+h_q{e}_q$ is a unit exactly when there is $k=k_p{e}_p+k_q{e}_q$ with $hk=\tilde{1}$. Component comparison gives $h_p{k}_p=1$ and $h_q{k}_q=1$ in $\mathbb{R}\left[t\right]$, so $h_p$ and $h_q$ are nonzero real constants. The converse is immediate from the inverse $c_p^{-1}{e}_p+c_q^{-1}{e}_q$. The divisibility assertion is proved in the same way: $b=ac$ holds in ${\mathcal{O}}_{\phi }\left[t\right]$ precisely when $b_p=a_p{c}_p$ and $b_q=a_q{c}_q$ hold in $\mathbb{R}\left[t\right]$.

The unimodular statement is the matrix version of the same argument. If $U\left(t\right)W\left(t\right)=W\left(t\right)U\left(t\right)=I$, comparison of the two channels gives polynomial inverses for $U_p\left(t\right)$ and $U_q\left(t\right)$. Conversely, channelwise polynomial inverses combine by (3.8), and multiplication is valid because the mixed idempotent products vanish. □

The next theorem is the polynomial normal form behind the whole paper. Its invariant factors are not ordinary real polynomials, and they are not arbitrary elements of a ring with zero divisors. They are paired real invariant factors joined by the two primitive idempotents. This is why the theorem produces intrinsic OPNs data even though its proof uses two classical Smith normal forms.

Theorem 3.3

(Idempotent Smith normal form and determinantal ideals). Let $F\left(t\right)\in M_{m,n}\left({\mathcal{O}}_{\phi }\left[t\right]\right)$ and write

$$F\left(t\right)=F_p\left(t\right)e_p+F_q\left(t\right)e_q.$$

Then there exist unimodular OPNs polynomial matrices $U\left(t\right)\in G{L}_m\left({\mathcal{O}}_{\phi }\left[t\right]\right)$ and $V\left(t\right)\in G{L}_n\left({\mathcal{O}}_{\phi }\left[t\right]\right)$ such that

$$U\left(t\right)F\left(t\right)V\left(t\right)=D\left(t\right),$$

(3.9)

where

$$D\left(t\right)=diag(d_1\left(t\right),\dots ,d_s\left(t\right),0,\dots ,0),s=min(m,n),$$

(3.10)

and

$$d_i\left(t\right)=f_i\left(t\right)e_p+g_i\left(t\right)e_q,f_i\left(t\right),g_i\left(t\right)\in \mathbb{R}\left[t\right].$$

(3.11)

The divisibility chains

$$f_i\left(t\right)\mid f_{i+1}\left(t\right),g_i\left(t\right)\mid g_{i+1}\left(t\right)$$

(3.12)

hold for all admissible i. The factors $d_i\left(t\right)$ are unique up to multiplication by units of ${\mathcal{O}}_{\phi }\left[t\right]$. If ${\Delta }_k\left(F\right)$ denotes the ideal of ${\mathcal{O}}_{\phi }\left[t\right]$ generated by all $k\times k$ minors of $F\left(t\right)$ and ${\Delta }_0\left(F\right)={\mathcal{O}}_{\phi }\left[t\right]$, then

$${\Delta }_k\left(F\right)=\left(d_1\left(t\right)d_2\left(t\right)\dots d_k\left(t\right)\right),1\le k\le s.$$

(3.13)

Equivalently,

$${\Delta }_k\left(F\right)=\left(f_1\dots f_k\right)e_p\oplus \left(g_1\dots g_k\right)e_q.$$

(3.14)

Proof. 

Since $\mathbb{R}\left[t\right]$ is a principal ideal domain, the ordinary Smith normal form applies separately to $F_p\left(t\right)$ and $F_q\left(t\right)$. Thus there exist unimodular real polynomial matrices $U_p,V_p,U_q,V_q$ such that

$$U_p{F}_p{V}_p=D_p,U_q{F}_q{V}_q=D_q,$$

where

$$D_p=diag(f_1,\dots ,f_s,0,\dots ,0),D_q=diag(g_1,\dots ,g_s,0,\dots ,0),$$

and the two real divisibility chains in (3.12) hold. Put

$$U=U_p{e}_p+U_q{e}_q,V=V_p{e}_p+V_q{e}_q.$$

By Theorem 3.2, U and V are unimodular over ${\mathcal{O}}_{\phi }\left[t\right]$. Multiplying in the two channels gives

$$UFV=U_p{F}_p{V}_p{e}_p+U_q{F}_q{V}_q{e}_q=D_p{e}_p+D_q{e}_q,$$

whose diagonal entries are $f_i{e}_p+g_i{e}_q$. This proves existence.

For uniqueness, let $D=diag\left(d_i\right)$ and $D^{\prime }=diag\left(d_i^{\prime }\right)$ be two OPNs Smith diagonals equivalent to F. Splitting the equivalence into the two polynomial channels gives two ordinary Smith forms for $F_p$ and two ordinary Smith forms for $F_q$. The uniqueness part of the real Smith theorem gives

$$f_i^{\prime }\left(t\right)=c_{p,i}{f}_i\left(t\right),g_i^{\prime }\left(t\right)=c_{q,i}{g}_i\left(t\right),$$

with nonzero real constants $c_{p,i}$ and $c_{q,i}$, with the usual zero-factor convention. Therefore

$$d_i^{\prime }\left(t\right)=(c_{p,i}{e}_p+c_{q,i}{e}_q)d_i\left(t\right),$$

and the multiplier is a unit by (3.7). This proves the stated uniqueness.

Unimodular row and column operations preserve determinantal ideals, so it is enough to compute the ideal for the Smith diagonal. A $k\times k$ minor of D is either zero or a product of k chosen diagonal entries. The divisibility chains imply that $d_1\dots d_k$ divides every nonzero such product, while the leading $k\times k$ principal minor is exactly $d_1\dots d_k$. Hence the minors generate the principal ideal (3.13). Expanding the product through the idempotents gives (3.14). □

The Smith theorem is equivalent to a classification of finite presentations over ${\mathcal{O}}_{\phi }\left[t\right]$. This form is deeper than the matrix statement because it identifies the module category controlled by the polynomial theory. The zero divisors of ${\mathcal{O}}_{\phi }\left[t\right]$ do not disappear; instead, they force every finite presentation to split into two ordinary PID presentations.

Theorem 3.4

(Finite presentation classification over OPNs polynomial modules). Let M be a finitely presented ${\mathcal{O}}_{\phi }\left[t\right]$-module. Then there are uniquely determined nonnegative integers $a,b$ and uniquely determined monic nonconstant real polynomial chains

$$f_1\left(t\right)\mid f_2\left(t\right)\mid \dots \mid f_r\left(t\right),g_1\left(t\right)\mid g_2\left(t\right)\mid \dots \mid g_s\left(t\right),$$

(3.15)

up to omission of unit factors, such that

$$M\cong \left(\mathbb{R}{\left[t\right]}^a\oplus {\displaystyle \underset{i=1}{\overset{r}{⨁}}}\mathbb{R}\left[t\right]/\left(f_i\left(t\right)\right)\right)e_p\oplus \left(\mathbb{R}{\left[t\right]}^b\oplus {\displaystyle \underset{j=1}{\overset{s}{⨁}}}\mathbb{R}\left[t\right]/\left(g_j\left(t\right)\right)\right)e_q.$$

(3.16)

Equivalently, finitely presented ${\mathcal{O}}_{\phi }\left[t\right]$-modules are classified by a pair of ordinary finitely generated $\mathbb{R}\left[t\right]$-module invariant data, one in the p-channel and one in the q-channel.

Proof. 

Since $e_p$ and $e_q$ are orthogonal idempotents with sum $\tilde{1}$, every ${\mathcal{O}}_{\phi }\left[t\right]$-module M decomposes as

$$M=e_{p}M\oplus e_{q}M.$$

The summand $e_{p}M$ is naturally a module over the p-channel ring $e_p{\mathcal{O}}_{\phi }\left[t\right]\cong \mathbb{R}\left[t\right]$, and $e_{q}M$ is naturally a module over the q-channel ring $e_q{\mathcal{O}}_{\phi }\left[t\right]\cong \mathbb{R}\left[t\right]$. If M has a finite presentation over ${\mathcal{O}}_{\phi }\left[t\right]$, applying $e_p$ and $e_q$ to that presentation gives finite presentations of $e_{p}M$ and $e_{q}M$ over the two copies of $\mathbb{R}\left[t\right]$.

The structure theorem for finitely generated modules over the PID $\mathbb{R}\left[t\right]$ gives decompositions

$$e_{p}M\cong \mathbb{R}{\left[t\right]}^a\oplus {\displaystyle \underset{i=1}{\overset{r}{⨁}}}\mathbb{R}\left[t\right]/\left(f_i\left(t\right)\right),e_{q}M\cong \mathbb{R}{\left[t\right]}^b\oplus {\displaystyle \underset{j=1}{\overset{s}{⨁}}}\mathbb{R}\left[t\right]/\left(g_j\left(t\right)\right),$$

where the displayed invariant chains are unique after taking the nonzero torsion factors to be monic and nonconstant. Recombining the two summands with the idempotents gives (3.16). Conversely, every module of the displayed form is finitely presented over ${\mathcal{O}}_{\phi }\left[t\right]$, because each summand is finitely presented over its corresponding PID channel and the two channels recombine through the product algebra. Uniqueness follows from the uniqueness of the PID invariant factors in each channel and from the uniqueness of the idempotent splitting $M=e_{p}M\oplus e_{q}M$. □

The finite-presentation theorem also reveals an obstruction that does not occur over a single polynomial PID. The projective-module background follows the usual module-theoretic viewpoint over product rings and PID channels [12,13]. Projective modules over ${\mathcal{O}}_{\phi }\left[t\right]$ carry two independent ranks. Equality of these ranks is exactly the condition for freeness.

Theorem 3.5

(Projective rank pairs and nonfree projectives over the OPNs polynomial ring). Let $R_{{\mathcal{O}}_{\phi }}={\mathcal{O}}_{\phi }\left[t\right]$. Every finitely generated projective $R_{{\mathcal{O}}_{\phi }}$-module P has a unique rank pair $(a,b)$ such that

$$P\cong \mathbb{R}{\left[t\right]}^a{e}_p\oplus \mathbb{R}{\left[t\right]}^b{e}_q.$$

(3.17)

Moreover, P is free as an $R_{{\mathcal{O}}_{\phi }}$-module if and only if $a=b$. In particular, $R_{{\mathcal{O}}_{\phi }}{e}_p$ and $R_{{\mathcal{O}}_{\phi }}{e}_q$ are finitely generated projective $R_{{\mathcal{O}}_{\phi }}$-modules which are not free.

Proof. 

The idempotent splitting of $R_{{\mathcal{O}}_{\phi }}$ gives

$$R_{{\mathcal{O}}_{\phi }}=\mathbb{R}\left[t\right]e_p\oplus \mathbb{R}\left[t\right]e_q.$$

For every $R_{{\mathcal{O}}_{\phi }}$-module P there is a corresponding decomposition

$$P=e_{p}P\oplus e_{q}P.$$

If P is finitely generated and projective over $R_{{\mathcal{O}}_{\phi }}$, then $e_{p}P$ and $e_{q}P$ are finitely generated projective modules over the two channel rings $\mathbb{R}\left[t\right]e_p\cong \mathbb{R}\left[t\right]$ and $\mathbb{R}\left[t\right]e_q\cong \mathbb{R}\left[t\right]$. Since $\mathbb{R}\left[t\right]$ is a principal ideal domain, every finitely generated projective module over $\mathbb{R}\left[t\right]$ is free. Thus there exist uniquely determined nonnegative integers $a,b$ such that

$$e_{p}P\cong \mathbb{R}{\left[t\right]}^a{e}_p,e_{q}P\cong \mathbb{R}{\left[t\right]}^b{e}_q.$$

Recombining the two idempotent summands gives (3.17). The uniqueness of $(a,b)$ follows from the uniqueness of ranks of free modules over the domain $\mathbb{R}\left[t\right]$ in the two channels.

If P is free of rank r over $R_{{\mathcal{O}}_{\phi }}$, then

$$P\cong R_{{\mathcal{O}}_{\phi }}^r=\mathbb{R}{\left[t\right]}^r{e}_p\oplus \mathbb{R}{\left[t\right]}^r{e}_q,$$

so its projective rank pair is $(r,r)$. Conversely, if $a=b=r$, then

$$\mathbb{R}{\left[t\right]}^r{e}_p\oplus \mathbb{R}{\left[t\right]}^r{e}_q\cong R_{{\mathcal{O}}_{\phi }}^r,$$

and P is free. Taking $P=R_{{\mathcal{O}}_{\phi }}{e}_p$ gives rank pair $(1,0)$, and taking $P=R_{{\mathcal{O}}_{\phi }}{e}_q$ gives rank pair $(0,1)$. Since neither pair is diagonal, neither module is free, although both are direct summands of the free module $R_{{\mathcal{O}}_{\phi }}$ and hence projective. □

4. Four-Pivot Rank-Pair Geometry

The constant-matrix theory is where the OPNs structure becomes visibly different from ordinary real matrix theory. A real matrix under row-column equivalence has only two local possibilities on a diagonal coordinate: a pivot or no pivot. An OPNs matrix has two spectral channels, so each diagonal coordinate can be active in both channels, only the p-channel, only the q-channel, or neither channel. This gives four pivot types and forces rank to become a pair rather than a single integer. The rank pair does more than classify normal forms: it controls stable direct sums, factorization through matrices, orbit geometry, and determinantal ideal chains.

Two matrices $A,B\in M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ are called equivalent if there exist $U\in G{L}_m\left({\mathcal{O}}_{\phi }\right)$ and $V\in G{L}_n\left({\mathcal{O}}_{\phi }\right)$ such that $B=UAV$. For $0\le k\le min(m,n)$, let $E_k^{m,n}$ be the $m\times n$ real matrix whose first k diagonal entries are 1 and whose remaining entries are 0; when the size is clear, write $E_k$.

Theorem 4.1

(Four-pivot rank-pair geometry). Fix nonnegative integers $m,n$ and let $s=min(m,n)$. The action

$$(U,V)·A=UAV,(U,V)\in G{L}_m\left({\mathcal{O}}_{\phi }\right)\times G{L}_n\left({\mathcal{O}}_{\phi }\right),$$

on $M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ has the following structure.

(1)

The orbits are indexed by rank pairs

$$(a,b)=(rank{A}_p,rank{A}_q),0\le a,b\le s.$$

(4.1)

The orbit of rank pair $(a,b)$ contains the four-pivot diagonal $D_{a,b}^{m,n}$ whose diagonal entries consist of

$$\underset{min(a,b)}{\underbrace{\tilde{1},\dots ,\tilde{1}}},\underset{{(a-b)}_{+}}{\underbrace{e_p,\dots ,e_p}},\underset{{(b-a)}_{+}}{\underbrace{e_q,\dots ,e_q}},\underset{s-max(a,b)}{\underbrace{\tilde{0},\dots ,\tilde{0}}},$$

(4.2)

with all off-diagonal entries equal to $\tilde{0}$ and with extra zero rows or columns when $m\ne n$.

(2)

For $A,B\in M_{m,n}\left({\mathcal{O}}_{\phi }\right)$, the following conditions are equivalent:

(a)

A and B are equivalent over ${\mathcal{O}}_{\phi }$;

(b)

$rank{A}_p=rank{B}_p$ and $rank{A}_q=rank{B}_q$;

(c)

A and B have the same four-pivot profile

$${\rho }_1=min(a,b),{\rho }_p={(a-b)}_{+},{\rho }_q={(b-a)}_{+},{\rho }_0=s-max(a,b).$$

(4.3)

(3)

The four local pivots

$$\tilde{1},e_p,e_q,\tilde{0}$$

(4.4)

are pairwise inequivalent. Consequently the rank pair

$${rank}_{\mathcal{O}}\left(A\right)=(rank{A}_p,rank{A}_q)$$

(4.5)

is the minimal complete rank-type invariant for OPNs matrix equivalence; no single integer-valued rank function that identifies two distinct rank pairs can classify all rectangular OPNs matrices under equivalence.

(4)

Let ${\mathcal{M}}_{{\mathcal{O}}_{\phi }}$ be the commutative monoid of stable equivalence classes of finite rectangular OPNs matrices under block direct sum, where zero source and zero target summands are ignored. Then the rank-pair map induces a monoid isomorphism

$${\mathcal{M}}_{{\mathcal{O}}_{\phi }}\cong {\mathbb{N}}^2.$$

(4.6)

Under this isomorphism,

$$\left[e_p\right]\mapsto (1,0),\left[e_q\right]\mapsto (0,1),\left[\tilde{1}\right]\mapsto (1,1),\left[\tilde{0}\right]\mapsto (0,0).$$

(5)

Let $A\in M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ and $B\in M_{r,u}\left({\mathcal{O}}_{\phi }\right)$. Then A factors through B, meaning that there exist OPNs matrices $C,D$ of compatible sizes with

$$A=CBD,$$

(4.7)

if and only if

$$rank{A}_p\le rank{B}_p,rank{A}_q\le rank{B}_q.$$

(4.8)

Thus the stable factorization preorder is exactly the product order on ${\mathbb{N}}^2$.

(6)

As a real manifold, the orbit

$${\mathcal{O}}_{a,b}^{m,n}=\{A\in M_{m,n}\left({\mathcal{O}}_{\phi }\right):rank{A}_p=a,rank{A}_q=b\}$$

has dimension

$${dim}_{\mathbb{R}}{\mathcal{O}}_{a,b}^{m,n}=a(m+n-a)+b(m+n-b).$$

(4.9)

Its Euclidean closure in $M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ is

$$\overline{{\mathcal{O}}_{a,b}^{m,n}}=\bigcup _{\begin{array}{c}0\le c\le a\\ 0\le d\le b\end{array}}{\mathcal{O}}_{c,d}^{m,n}.$$

(4.10)

(7)

Let $J_k\left(A\right)$ be the ideal of ${\mathcal{O}}_{\phi }$ generated by the $k\times k$ minors of A, with $J_0\left(A\right)={\mathcal{O}}_{\phi }$. For an ideal $J\subseteq {\mathcal{O}}_{\phi }$, write

$${\pi }_p\left(J\right)=\{p_{\alpha }:\alpha \in J\},{\pi }_q\left(J\right)=\{q_{\alpha }:\alpha \in J\}.$$

Then the rank pair of A is recovered by

$$rank{A}_p=max\{k:{\pi }_p\left(J_k\left(A\right)\right)\ne 0\},rank{A}_q=max\{k:{\pi }_q\left(J_k\left(A\right)\right)\ne 0\}.$$

(4.11)

Consequently the determinantal ideal chain is an intrinsic complete invariant for matrix equivalence over ${\mathcal{O}}_{\phi }$.

Proof. 

The matrix splitting theorem identifies $M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ with $M_{m,n}\left(\mathbb{R}\right)\times M_{m,n}\left(\mathbb{R}\right)$ by sending A to $(A_p,A_q)$. Under this identification, the group $G{L}_m\left({\mathcal{O}}_{\phi }\right)\times G{L}_n\left({\mathcal{O}}_{\phi }\right)$ becomes the product of the two real left-right equivalence groups in the two spectral channels. Therefore two OPNs matrices lie in the same orbit precisely when their p-components have the same real rank and their q-components have the same real rank. This proves that the orbits are indexed by rank pairs $(a,b)$.

For a real matrix of rank a, ordinary row-column equivalence reduces it to $E_a^{m,n}$. Applying this in the p-channel and the q-channel independently gives real invertible matrices $U_p,V_p,U_q,V_q$ such that

$$U_p{A}_p{V}_p=E_a,U_q{A}_q{V}_q=E_b.$$

Recombining

$$U=U_p{e}_p+U_q{e}_q,V=V_p{e}_p+V_q{e}_q,$$

gives invertible OPNs matrices and

$$UAV=E_a{e}_p+E_b{e}_q.$$

At each diagonal coordinate, the possible pairs of real diagonal entries are $(1,1)$, $(1,0)$, $(0,1)$, and $(0,0)$, which correspond respectively to $\tilde{1},e_p,e_q,\tilde{0}$. Counting the overlap of the first a and first b diagonal positions gives the diagonal in (4.2). Conversely, equivalence over ${\mathcal{O}}_{\phi }$ splits into equivalence in both real channels, so the two real ranks are preserved. This proves the normal form and the equivalence of the three classification conditions. The pivot profile is computed from $(a,b)$ by (4.3), while the rank pair is recovered from the profile by

$$a={\rho }_1+{\rho }_p,b={\rho }_1+{\rho }_q.$$

The four $1\times 1$ pivots have rank pairs

$$(1,1),(1,0),(0,1),(0,0),$$

respectively. Since matrix equivalence preserves the two real ranks, these four local types are pairwise inequivalent. The classification just proved shows that the ordered pair $(rank{A}_p,rank{A}_q)$ is complete. Any rank-type invariant that collapses two distinct pairs cannot distinguish the corresponding four-pivot normal forms, and hence cannot classify OPNs matrices up to equivalence. Thus the irreducible information is the rank pair, or equivalently the four-pivot profile.

Block direct sum adds real ranks in each channel:

$$rank{(A\oplus B)}_p=rank{A}_p+rank{B}_p,rank{(A\oplus B)}_q=rank{A}_q+rank{B}_q.$$

Hence the rank-pair map is a homomorphism from the stable monoid of OPNs matrices to ${\mathbb{N}}^2$. Surjectivity follows because the direct sum of a copies of $e_p$ and b copies of $e_q$ has rank pair $(a,b)$. Injectivity follows from the normal-form classification: two stable classes with the same rank pair have the same four-pivot stable normal form after zero source and target summands are removed. This proves (4.6), and the displayed images of $e_p,e_q,\tilde{1},\tilde{0}$ are immediate from their rank pairs.

For the factorization statement, suppose first that $A=CBD$. Passing to spectral components gives

$$A_p=C_p{B}_p{D}_p,A_q=C_q{B}_q{D}_q.$$

The ordinary real rank inequality gives

$$rank{A}_p\le rank{B}_p,rank{A}_q\le rank{B}_q.$$

Conversely, assume the two inequalities in (49). Over a field, a matrix X factors through a matrix Y if and only if $rankX\le rankY$. Applying this real fact in the two spectral channels, there exist real matrices $C_p,D_p,C_q,D_q$ such that

$$A_p=C_p{B}_p{D}_p,A_q=C_q{B}_q{D}_q.$$

Set

$$C=C_p{e}_p+C_q{e}_q,D=D_p{e}_p+D_q{e}_q.$$

The matrix transfer principle gives

$$CBD=C_p{B}_p{D}_p{e}_p+C_q{B}_q{D}_q{e}_q=A_p{e}_p+A_q{e}_q=A.$$

This proves the factorization criterion and identifies the stable factorization preorder with the product order on ${\mathbb{N}}^2$.

The orbit dimension and closure formula follow from the same product description, now using the standard geometry of real rank strata. The real rank-a stratum in $M_{m,n}\left(\mathbb{R}\right)$ is a smooth manifold of dimension $a(m+n-a)$. Since the OPNs rank-pair orbit is the product of the real rank-a and real rank-b strata, its real dimension is the sum

$$a(m+n-a)+b(m+n-b),$$

which proves (4.9). The Euclidean closure of the real rank-a stratum consists of all real matrices of rank at most a. Taking the product of the two closure relations gives (4.10).

It remains to connect the same rank-pair geometry with determinantal ideals. A $k\times k$ minor of A has spectral components equal to the corresponding $k\times k$ minors of $A_p$ and $A_q$. Hence $J_k\left(A\right)$ has nonzero p-projection if and only if $A_p$ has at least one nonzero $k\times k$ minor, which is equivalent to $k\le rank{A}_p$. The same argument in the q-channel shows that $J_k\left(A\right)$ has nonzero q-projection if and only if $k\le rank{A}_q$. The two formulas in (4.11) follow. Since the rank pair classifies matrix equivalence, the determinantal ideal chain is an intrinsic complete invariant for equivalence. □

The theorem shows that four-pivot form is not merely a convenient diagonal notation. It is the common normal form for equivalence, stable addition, factorization, orbit closure, and determinantal recovery. In particular, the objects $e_p$ and $e_q$ behave as two independent primitive rank generators, while $11$ represents their simultaneous occurrence in one coordinate.

5. Modules, Determinantal Ideals, and Linear Systems

In this section a matrix $A\in M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ is regarded as an ${\mathcal{O}}_{\phi }$-module homomorphism $A:{\mathcal{O}}_{\phi }^n\to {\mathcal{O}}_{\phi }^m$. The passage from a diagonal normal form to kernels and cokernels is not merely a bookkeeping exercise. Because $e_p$ and $e_q$ are zero divisors, a one-channel pivot has both an image contribution and a complementary kernel or cokernel contribution. This is the module-theoretic meaning of the four-pivot normal form.

Theorem 5.1

(Finite OPNs module classification and exact splitting). Every finitely generated ${\mathcal{O}}_{\phi }$-module M decomposes as

$$M=e_{p}M\oplus e_{q}M,$$

(5.1)

where $e_{p}M$ and $e_{q}M$ are finite-dimensional real vector spaces. Hence there are uniquely determined nonnegative integers $a,b$ such that

$$M\cong {\left({\mathcal{O}}_{\phi }{e}_p\right)}^a\oplus {\left({\mathcal{O}}_{\phi }{e}_q\right)}^b.$$

(5.2)

Equivalently, if $r=min(a,b)$, then

$$M\cong {\mathcal{O}}_{\phi }^r\oplus {\left({\mathcal{O}}_{\phi }{e}_p\right)}^{a-r}\oplus {\left({\mathcal{O}}_{\phi }{e}_q\right)}^{b-r}.$$

(5.3)

Moreover, every short exact sequence of finitely generated ${\mathcal{O}}_{\phi }$-modules

$$0⟶M^{\prime }⟶M⟶M^{\prime \prime }⟶0$$

(5.4)

splits. Consequently, every submodule of a finitely generated ${\mathcal{O}}_{\phi }$-module is a direct summand.

Proof. 

Since $\tilde{1}=e_p+e_q$ and $e_p{e}_q=\tilde{0}$, every element $x\in M$ decomposes as

$$x=e_{p}x+e_{q}x.$$

The intersection of $e_{p}M$ and $e_{q}M$ is zero, because if $x=e_{p}u=e_{q}v$, then applying $e_p$ gives $x=e_{p}x=e_p{e}_{q}v=0$. This proves (5.1). The summand $e_{p}M$ is a vector space over ${\mathcal{O}}_{\phi }{e}_p\cong \mathbb{R}$, and $e_{q}M$ is a vector space over ${\mathcal{O}}_{\phi }{e}_q\cong \mathbb{R}$. Finite generation over ${\mathcal{O}}_{\phi }$ implies finite dimensionality over these two real fields. Therefore $e_{p}M\cong {\left({\mathcal{O}}_{\phi }{e}_p\right)}^a$ and $e_{q}M\cong {\left({\mathcal{O}}_{\phi }{e}_q\right)}^b$ for uniquely determined dimensions $a,b$, which gives (5.2). Pairing $r=min(a,b)$ copies of ${\mathcal{O}}_{\phi }{e}_p$ with ${\mathcal{O}}_{\phi }{e}_q$ gives ${\mathcal{O}}_{\phi }^r$, and the unpaired summands give (5.3).

For a short exact sequence as in (5.4), applying $e_p$ and $e_q$ gives two short exact sequences of finite-dimensional real vector spaces. Both split over $\mathbb{R}$. Choosing real splittings in the two channels and recombining them with $e_p$ and $e_q$ gives an ${\mathcal{O}}_{\phi }$-linear splitting of the original sequence. If $N\subseteq M$ is a submodule, the exact sequence

$$0⟶N⟶M⟶M/N⟶0$$

splits, so N is a direct summand. □

Theorem 5.2

(Canonical module and determinantal structure). Let $A\in M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ and let $a=rank{A}_p$ and $b=rank{A}_q$. Then

$$\begin{array}{cc}\hfill kerA& \cong {\left({\mathcal{O}}_{\phi }{e}_q\right)}^{{(a-b)}_{+}}\oplus {\left({\mathcal{O}}_{\phi }{e}_p\right)}^{{(b-a)}_{+}}\oplus {\mathcal{O}}_{\phi }^{n-max(a,b)},\hfill \\ \hfill imA& \cong {\mathcal{O}}_{\phi }^{min(a,b)}\oplus {\left({\mathcal{O}}_{\phi }{e}_p\right)}^{{(a-b)}_{+}}\oplus {\left({\mathcal{O}}_{\phi }{e}_q\right)}^{{(b-a)}_{+}},\hfill \\ \hfill cokerA& \cong {\left({\mathcal{O}}_{\phi }{e}_q\right)}^{{(a-b)}_{+}}\oplus {\left({\mathcal{O}}_{\phi }{e}_p\right)}^{{(b-a)}_{+}}\oplus {\mathcal{O}}_{\phi }^{m-max(a,b)}.\hfill \end{array}$$

(5.5)

The two-channel rank-nullity identity is

$${rank}_{\mathcal{O}}\left(A\right)+{nullity}_{\mathcal{O}}\left(A\right)=(n,n),$$

(5.6)

where

$${rank}_{\mathcal{O}}\left(A\right)=(rank{A}_p,rank{A}_q),{nullity}_{\mathcal{O}}\left(A\right)=(dimker{A}_p,dimker{A}_q).$$

For $1\le k\le min(m,n)$, let $J_k\left(A\right)$ be the ideal of ${\mathcal{O}}_{\phi }$ generated by the $k\times k$ minors of A, with $J_0\left(A\right)={\mathcal{O}}_{\phi }$. Then

$$J_k\left(A\right)=\left\{\begin{array}{cc}{\mathcal{O}}_{\phi },\hfill & k\le a\mathrm{and}k\le b,\hfill \\ {\mathrm{I}}_q,\hfill & k\le a\mathrm{and}k>b,\hfill \\ {\mathrm{I}}_p,\hfill & k>a\mathrm{and}k\le b,\hfill \\ \left\{\tilde{0}\right\},\hfill & k>a\mathrm{and}k>b.\hfill \end{array}\right.$$

(5.7)

Proof. 

The one-coordinate maps determine the module structure of a four-pivot diagonal. If $x=x_p{e}_p+x_q{e}_q$, then $e_{p}x=x_p{e}_p$. Therefore multiplication by $e_p$ has kernel ${\mathcal{O}}_{\phi }{e}_q$, image ${\mathcal{O}}_{\phi }{e}_p$, and cokernel naturally isomorphic to ${\mathcal{O}}_{\phi }{e}_q$. Similarly, multiplication by $e_q$ has kernel ${\mathcal{O}}_{\phi }{e}_p$, image ${\mathcal{O}}_{\phi }{e}_q$, and cokernel isomorphic to ${\mathcal{O}}_{\phi }{e}_p$. Multiplication by $\tilde{1}$ has zero kernel and cokernel and image ${\mathcal{O}}_{\phi }$, while multiplication by $\tilde{0}$ has kernel ${\mathcal{O}}_{\phi }$, image 0, and cokernel ${\mathcal{O}}_{\phi }$.

By Theorem 4.1, there are invertible matrices U and V with $UAV=D_{a,b}^{m,n}$. Multiplication by V is a domain automorphism and multiplication by U is a codomain automorphism, so A and $D_{a,b}^{m,n}$ have isomorphic kernels, images, and cokernels. The diagonal map $D_{a,b}^{m,n}$ is a direct sum of the one-coordinate maps just computed, together with extra zero rows or columns when $m\ne n$. A $\tilde{1}$-pivot contributes one copy of ${\mathcal{O}}_{\phi }$ to the image and no kernel or cokernel; an $e_p$-pivot contributes ${\mathcal{O}}_{\phi }{e}_q$ to the kernel, ${\mathcal{O}}_{\phi }{e}_p$ to the image, and ${\mathcal{O}}_{\phi }{e}_q$ to the cokernel; an $e_q$-pivot contributes ${\mathcal{O}}_{\phi }{e}_p$ to the kernel, ${\mathcal{O}}_{\phi }{e}_q$ to the image, and ${\mathcal{O}}_{\phi }{e}_p$ to the cokernel. The fully zero part and extra domain coordinates contribute free kernel summands, while extra target coordinates contribute free cokernel summands. Counting these contributions gives (5.5). The rank-nullity identity follows either from the displayed kernel formula by reading the two channels or directly from the real identities $rank{A}_p+dimker{A}_p=n$ and $rank{A}_q+dimker{A}_q=n$.

For determinantal ideals, a $k\times k$ minor of A has spectral components equal to the corresponding minors of $A_p$ and $A_q$. If $k\le a$ and $k\le b$, then both real matrices have a nonzero $k\times k$ minor, so the generated ideal has nonzero projection in both channels and must be ${\mathcal{O}}_{\phi }$. If $k\le a$ and $k>b$, then some p-minor is nonzero while every q-minor is zero; all OPNs minors lie in ${\mathrm{I}}_q=\mathbb{R}{e}_p$, and at least one has nonzero p-component, so the ideal is ${\mathrm{I}}_q$. The case $k>a$ and $k\le b$ gives ${\mathrm{I}}_p$ by the same argument with the two channels interchanged. If $k>a$ and $k>b$, all such minors vanish in both channels, so $J_k\left(A\right)=\left\{\tilde{0}\right\}$. □

The same diagonal analysis gives the correct form of linear-system solvability. The usual real Rouché–Capelli theorem survives, but it must be imposed simultaneously in the p- and q-channels. After four-pivot reduction, the equations separate into coordinates that fix both spectral components, coordinates that fix only one component, and coordinates that impose compatibility on the right-hand side.

Theorem 5.3

(Linear systems and four-pivot Rouché–Capelli theorem). Let $A\in M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ and $b\in {\mathcal{O}}_{\phi }^m$. The system

$$Ax=b$$

(5.8)

has a solution in ${\mathcal{O}}_{\phi }^n$ precisely when the two real systems

$$A_p{x}_p=b_p,A_q{x}_q=b_q$$

(5.9)

have real solutions. Equivalently,

$$rank{A}_p=rank\left(A_p\right|b_p),rank{A}_q=rank\left(A_q\right|b_q).$$

(5.10)

When solutions exist, the solution set is $x_0+kerA$ for any particular solution $x_0$. If $UAV=D_{a,b}^{m,n}$ and one sets $c=Ub$ and $y=V^{-1}x$, then (5.8) is equivalent to $D_{a,b}^{m,n}y=c$. In this reduced system, a $11$-pivot fixes both spectral components of the corresponding unknown, an $e_p$-pivot fixes only the p-component, an $e_q$-pivot fixes only the q-component, and a $\tilde{0}$-pivot imposes vanishing of the corresponding right-hand side coordinate while leaving the corresponding domain coordinate free when such a coordinate is present.

Proof. 

Write $x=x_p{e}_p+x_q{e}_q$ and $b=b_p{e}_p+b_q{e}_q$. By the matrix product splitting,

$$Ax=\left(A_p{x}_p\right)e_p+\left(A_q{x}_q\right)e_q.$$

Thus $Ax=b$ holds exactly when the two real equations in (5.9) hold. Applying the ordinary real Rouché–Capelli theorem to the two channel systems gives (5.10). If $x_0$ is one solution, then x is another solution precisely when $A(x-x_0)=\tilde{0}$, which means $x-x_0\in kerA$.

For the four-pivot interpretation, multiply $Ax=b$ on the left by U and substitute $x=Vy$. The resulting equation is $D_{a,b}^{m,n}y=c$. Each diagonal coordinate is then one of the four one-coordinate equations $\tilde{1}{y}_i=c_i$, $e_p{y}_i=c_i$, $e_q{y}_i=c_i$, or $\tilde{0}{y}_i=c_i$. The assertions follow from the multiplication rules used in the proof of Theorem 5.2. For instance, $e_p{y}_i=c_i$ is solvable exactly when $c_i\in {\mathcal{O}}_{\phi }{e}_p$; in that case the p-component of $y_i$ is fixed and the q-component is free. □

6. Determinants, Adjugates, and Inverse Matrices

For square matrices, the determinant retains its familiar formal definition by the Leibniz formula. The difference is in its value: it is an OPNs element whose two spectral coordinates are the two ordinary real determinants. Consequently invertibility is not detected by a single nonzero scalar determinant, but by the unit condition in the scalar OPNs algebra, equivalently by nonvanishing in both channels.

Theorem 6.1

(Determinant, adjugate, inverse, and Cramer theorem). For $A=\left({\alpha }_{ij}\right)\in M_n\left({\mathcal{O}}_{\phi }\right)$, define

$${det}_{\mathcal{O}}\left(A\right)=\sum _{\sigma \in S_n}sgn\left(\sigma \right){\alpha }_{1\sigma \left(1\right)}{\alpha }_{2\sigma \left(2\right)}\dots {\alpha }_{n\sigma \left(n\right)}.$$

(6.1)

Then

$$S_{\phi }\left({det}_{\mathcal{O}}\left(A\right)\right)=(det{A}_p,det{A}_q).$$

(6.2)

The adjugate matrix defined by the usual cofactor formula satisfies

$$A{adj}_{\mathcal{O}}\left(A\right)={adj}_{\mathcal{O}}\left(A\right)A={det}_{\mathcal{O}}\left(A\right)I.$$

(6.3)

For $A\in M_n\left({\mathcal{O}}_{\phi }\right)$, the following statements are equivalent:

(1)

A is invertible over ${\mathcal{O}}_{\phi }$;

(2)

$A_p$ and $A_q$ are invertible over $\mathbb{R}$;

(3)

${det}_{\mathcal{O}}\left(A\right)$ is a unit of ${\mathcal{O}}_{\phi }$;

(4)

$det{A}_p\ne 0$ and $det{A}_q\ne 0$.

When these conditions hold,

$$A^{-1}=A_p^{-1}{e}_p+A_q^{-1}{e}_q={\left({det}_{\mathcal{O}}\left(A\right)\right)}^{-1}{adj}_{\mathcal{O}}\left(A\right).$$

(6.4)

If $b\in {\mathcal{O}}_{\phi }^n$, the unique solution of $Ax=b$ is given by

$$x_i={\displaystyle \frac{{det}_{\mathcal{O}}\left(A_i\left(b\right)\right)}{{det}_{\mathcal{O}}\left(A\right)}},i=1,\dots ,n,$$

(6.5)

where $A_i\left(b\right)$ is obtained from A by replacing its i-th column by b.

Proof. 

Applying $S_{\phi }$ to the Leibniz formula (6.1) preserves addition, multiplication, and real signs. The p-component is therefore the ordinary Leibniz formula for $det{A}_p$, and the q-component is the ordinary Leibniz formula for $det{A}_q$. This proves (6.2).

The adjugate identity is valid over any commutative ring, and here it can be checked directly. The $(i,j)$ entry of $A{adj}_{\mathcal{O}}\left(A\right)$ is

$${\displaystyle \sum _{k=1}^n}{\alpha }_{ik}{C}_{jk},$$

where $C_{jk}$ is the $(j,k)$ cofactor. If $i=j$, this is the cofactor expansion of ${det}_{\mathcal{O}}\left(A\right)$ along row i. If $i\ne j$, it is the determinant of the matrix obtained from A by replacing row j with row i, expanded along row j; that matrix has two equal rows and determinant $\tilde{0}$. Hence $A{adj}_{\mathcal{O}}\left(A\right)={det}_{\mathcal{O}}\left(A\right)I$. The identity on the other side is the analogous column expansion.

The equivalence of invertibility of A and invertibility of $A_p,A_q$ is Theorem 3.1. The equivalence with nonzero real determinants is the ordinary determinant criterion in the two channels. By (6.2) and the scalar unit criterion, ${det}_{\mathcal{O}}\left(A\right)$ is a unit precisely when $det{A}_p$ and $det{A}_q$ are both nonzero. This proves the four equivalent conditions. Formula (6.4) follows first from the channelwise inverse and then from (6.3) after multiplying by the unit ${\left({det}_{\mathcal{O}}\left(A\right)\right)}^{-1}$.

For Cramer’s rule, ${det}_{\mathcal{O}}\left(A\right)$ is a unit under the equivalent conditions above. Taking $S_{\phi }$ of (6.5), the p-component is the ordinary Cramer formula for $A_p{x}_p=b_p$, and the q-component is the ordinary Cramer formula for $A_q{x}_q=b_q$. Injectivity of the spectral map gives the OPNs formula. □

7. Characteristic Polynomials, Invariant Factors, and Similarity

The preceding sections classify matrices under row-column equivalence. Square matrices also carry similarity invariants, and these are controlled by the polynomial matrix $tI-A$. This is where the polynomial Smith theorem reenters: invariant factors for $tI-A$ package the rational canonical data of the two real channels into OPNs polynomial invariants.

For $A\in M_n\left({\mathcal{O}}_{\phi }\right)$ define the characteristic polynomial by

$${\chi }_A\left(t\right)={det}_{\mathcal{O}}(tI-A),$$

(7.1)

where t is a central real indeterminate.

Theorem 7.1

(Characteristic and minimal polynomial theorem). For every $A\in M_n\left({\mathcal{O}}_{\phi }\right)$,

$${\chi }_A\left(t\right)={\chi }_{A_p}\left(t\right)e_p+{\chi }_{A_q}\left(t\right)e_q,$$

(7.2)

where ${\chi }_{A_p}\left(t\right)=det(tI-A_p)$ and ${\chi }_{A_q}\left(t\right)=det(tI-A_q)$. The Cayley–Hamilton identity holds:

$${\chi }_A\left(A\right)=0.$$

(7.3)

Let $m_{A_p}\left(t\right)$ and $m_{A_q}\left(t\right)$ be the ordinary real minimal polynomials of $A_p$ and $A_q$. Among real-coefficient polynomials $f\left(t\right)\in \mathbb{R}\left[t\right]$, the monic polynomial of least degree satisfying $f\left(A\right)=0$ is

$$lcm(m_{A_p},m_{A_q}).$$

(7.4)

If coefficients in ${\mathcal{O}}_{\phi }$ are allowed, the channelwise minimal polynomial is

$$m_A^{\mathcal{O}}\left(t\right)=m_{A_p}\left(t\right)e_p+m_{A_q}\left(t\right)e_q,$$

(7.5)

and it satisfies $m_A^{\mathcal{O}}\left(A\right)=0$.

Proof. 

The matrix $tI-A$ has spectral components $tI-A_p$ and $tI-A_q$. Applying the determinant splitting from Theorem 6.1 gives (7.2). Evaluating at $A=A_p{e}_p+A_q{e}_q$ and using the matrix transfer principle gives

$${\chi }_A\left(A\right)={\chi }_{A_p}\left(A_p\right)e_p+{\chi }_{A_q}\left(A_q\right)e_q.$$

The ordinary Cayley–Hamilton theorem over $\mathbb{R}$ makes both real terms vanish, proving (7.3).

If $f\left(t\right)\in \mathbb{R}\left[t\right]$, then real coefficients act in both channels in the same way, so

$$f\left(A\right)=f\left(A_p\right)e_p+f\left(A_q\right)e_q.$$

Thus $f\left(A\right)=0$ precisely when $f\left(A_p\right)=0$ and $f\left(A_q\right)=0$. This means that $m_{A_p}$ and $m_{A_q}$ both divide f in $\mathbb{R}\left[t\right]$, and the least monic real polynomial with that property is their least common multiple. For OPNs coefficients, evaluation of (7.5) gives $m_{A_p}\left(A_p\right)e_p+m_{A_q}\left(A_q\right)e_q=0$. Minimality is channelwise: lowering either channel polynomial would contradict the corresponding real minimality. □

The polynomial Smith theorem gives intrinsic invariant factors for a matrix. Apply Theorem 3.3 to $tI-A$. If the nonzero diagonal entries are

$$d_i^A\left(t\right)=f_i^A\left(t\right)e_p+g_i^A\left(t\right)e_q,$$

then these are called the OPNs invariant factors of A.

Theorem 7.2

(Similarity and invariant-factor classification). For $A,B\in M_n\left({\mathcal{O}}_{\phi }\right)$, the following conditions are equivalent:

(1)

A and B are similar over ${\mathcal{O}}_{\phi }$;

(2)

$A_p$ is similar to $B_p$ over $\mathbb{R}$ and $A_q$ is similar to $B_q$ over $\mathbb{R}$;

(3)

the OPNs invariant factors of A and B agree up to multiplication by units in ${\mathcal{O}}_{\phi }\left[t\right]$;

(4)

the ordinary rational canonical data agree in both real channels.

Moreover, A is diagonalizable over ${\mathcal{O}}_{\phi }$ precisely when $A_p$ and $A_q$ are both diagonalizable over $\mathbb{R}$. Over any field extension where real Jordan forms exist, the Jordan data of A are the ordered pair of the Jordan data of $A_p$ and $A_q$.

Proof. 

If $B=T^{-1}AT$ with $T=T_p{e}_p+T_q{e}_q\in G{L}_n\left({\mathcal{O}}_{\phi }\right)$, then Theorem 3.1 gives

$$B_p=T_p^{-1}{A}_p{T}_p,B_q=T_q^{-1}{A}_q{T}_q.$$

Thus similarity over ${\mathcal{O}}_{\phi }$ implies similarity in both real channels. Conversely, if such real similarities are given by $T_p$ and $T_q$, then $T=T_p{e}_p+T_q{e}_q$ is invertible over ${\mathcal{O}}_{\phi }$ and the same componentwise multiplication gives $B=T^{-1}AT$.

Ordinary rational canonical theory over $\mathbb{R}$ says that real similarity is equivalent to equality of ordinary invariant factors, or equivalently equality of rational canonical data. The OPNs invariant factors of A are obtained by taking the Smith invariant factors of $tI-A_p$ and $tI-A_q$ and recombining the corresponding real factors as $f_i^A{e}_p+g_i^A{e}_q$. Therefore equality of OPNs invariant factors up to OPNs polynomial units is precisely equality of the two real invariant-factor lists up to nonzero real constants in the two channels. This proves the equivalence of all four conditions.

Diagonalization and Jordan data are special cases of the same similarity splitting. A diagonalization $A=TD{T}^{-1}$ over ${\mathcal{O}}_{\phi }$ splits into diagonalizations of $A_p$ and $A_q$. Conversely, diagonalizations of the two real components combine through $T=T_p{e}_p+T_q{e}_q$ and $D=D_p{e}_p+D_q{e}_q$. The Jordan statement is identical after replacing diagonal forms by the corresponding Jordan normal forms over a field extension where they exist. □

A useful strengthening of the similarity theorem is the channelwise description of commutants. For $A\in M_n\left({\mathcal{O}}_{\phi }\right)$ define

$${Cent}_{{\mathcal{O}}_{\phi }}\left(A\right)=\{X\in M_n\left({\mathcal{O}}_{\phi }\right):XA=AX\}.$$

Then

$${Cent}_{{\mathcal{O}}_{\phi }}\left(A\right)={Cent}_{\mathbb{R}}\left(A_p\right)e_p\oplus {Cent}_{\mathbb{R}}\left(A_q\right)e_q.$$

(7.6)

Indeed, $XA=AX$ is equivalent to $X_p{A}_p=A_p{X}_p$ and $X_q{A}_q=A_q{X}_q$. Thus the size of the commutant, cyclicity criteria, and all similarity invariants detected by centralizers are also two-channel invariants.

Similarity classification is channelwise, but eigenvalue language requires an additional convention. Because ${\mathcal{O}}_{\phi }$ has zero divisors, eigenvectors supported in only one channel produce degenerate spectral phenomena: such vectors can satisfy an eigenvalue equation while testing only half of the matrix. The regular spectral theory excludes this one-channel collapse by requiring both components of the eigenvector to be nonzero. Thus $\lambda \in {\mathcal{O}}_{\phi }$ is called a regular eigenvalue of A if there exists $v=v_p{e}_p+v_q{e}_q\in {\mathcal{O}}_{\phi }^n$ such that

$$Av=\lambda v,v_p\ne 0,v_q\ne 0.$$

Theorem 7.3

(Regular spectrum and spectral rectangles). For $A\in M_n\left({\mathcal{O}}_{\phi }\right)$,

$${Spec}_{reg}\left(A\right)=S_{\phi }^{-1}\left(Spec\left(A_p\right)\times Spec\left(A_q\right)\right).$$

(7.7)

Moreover, if $S_{\phi }\left(\lambda \right)=({\lambda }_p,{\lambda }_q)$, then the regular eigenmodule associated with λ is

$$E_{\mathcal{O}}(A,\lambda )=E(A_p,{\lambda }_p)e_p\oplus E(A_q,{\lambda }_q)e_q,$$

(7.8)

with both real eigenspaces required to be nonzero for regularity.

Proof. 

Let $Av=\lambda v$ with $v=v_p{e}_p+v_q{e}_q$ and $S_{\phi }\left(\lambda \right)=({\lambda }_p,{\lambda }_q)$. Taking spectral components gives

$$A_p{v}_p={\lambda }_p{v}_p,A_q{v}_q={\lambda }_q{v}_q.$$

(7.9)

If $v_p\ne 0$ and $v_q\ne 0$, then ${\lambda }_p\in Spec\left(A_p\right)$ and ${\lambda }_q\in Spec\left(A_q\right)$. Conversely, if ${\lambda }_p\in Spec\left(A_p\right)$ and ${\lambda }_q\in Spec\left(A_q\right)$, choose nonzero real eigenvectors $v_p$ and $v_q$ satisfying (7.9). With $v=v_p{e}_p+v_q{e}_q$ and $\lambda =S_{\phi }^{-1}({\lambda }_p,{\lambda }_q)$, one obtains

$$Av=\left(A_p{v}_p\right)e_p+\left(A_q{v}_q\right)e_q={\lambda }_p{v}_p{e}_p+{\lambda }_q{v}_q{e}_q=\lambda v.$$

Both components of v are nonzero, so $\lambda$ is regular. This proves (7.7), and the eigenspace formula (7.8) is exactly the same componentwise equation written for all eigenvectors. □

8. Symmetric Matrices, Congruence, and Inertia

The row-column normal form and similarity classification concern arbitrary matrices. For symmetric matrices the natural equivalence relation is congruence, and the classical reference point is Sylvester inertia theory for real symmetric forms [3,4]. Since transpose also splits channelwise, congruence over OPNs becomes two independent real congruence problems, and Sylvester inertia becomes a double inertia invariant.

A matrix $A\in M_n\left({\mathcal{O}}_{\phi }\right)$ is symmetric if $A^{\mathrm{T}}=A$. Two symmetric matrices A and B are congruent over ${\mathcal{O}}_{\phi }$ if there exists $C\in G{L}_n\left({\mathcal{O}}_{\phi }\right)$ such that

$$B=C^{\mathrm{T}}AC.$$

For a symmetric $A=A_p{e}_p+A_q{e}_q$, define its double inertia by

$${In}_{\mathcal{O}}\left(A\right)=\left((n_p^{+},n_p^{-},n_p^0),(n_q^{+},n_q^{-},n_q^0)\right),$$

(8.1)

where the two triples are the ordinary real inertias of $A_p$ and $A_q$.

Theorem 8.1

(Symmetric congruence and double inertia). Let $A,B\in M_n\left({\mathcal{O}}_{\phi }\right)$ be symmetric. Then A and B are congruent over ${\mathcal{O}}_{\phi }$ precisely when

$${In}_{\mathcal{O}}\left(A\right)={In}_{\mathcal{O}}\left(B\right).$$

(8.2)

Proof. 

If $B=C^{\mathrm{T}}AC$ with $C=C_p{e}_p+C_q{e}_q\in G{L}_n\left({\mathcal{O}}_{\phi }\right)$, then component comparison gives

$$B_p=C_p^{\mathrm{T}}{A}_p{C}_p,B_q=C_q^{\mathrm{T}}{A}_q{C}_q.$$

The ordinary real Sylvester law of inertia gives equality of the two real inertias in each channel, hence (8.2). Conversely, if the double inertias are equal, the real Sylvester law gives real invertible matrices $C_p$ and $C_q$ with

$$B_p=C_p^{\mathrm{T}}{A}_p{C}_p,B_q=C_q^{\mathrm{T}}{A}_q{C}_q.$$

Putting $C=C_p{e}_p+C_q{e}_q$ gives an invertible OPNs matrix, and multiplication in the two channels yields $C^{\mathrm{T}}AC=B$. □

A symmetric matrix $A\in M_n\left({\mathcal{O}}_{\phi }\right)$ is spectrally positive semidefinite if $A_p⪰0$ and $A_q⪰0$; this is the channelwise analogue of the standard real positive-semidefinite cone [3,14]. It is spectrally positive definite if $A_p\succ 0$ and $A_q\succ 0$. This definition is deliberately spectral rather than based on an order in normalized coordinates: positivity is a quadratic-form property in each real channel, and only then is it transported back to OPNs.

Theorem 8.2

(Spectral positivity, Cholesky factors, and square roots). A symmetric matrix $A\in M_n\left({\mathcal{O}}_{\phi }\right)$ is spectrally positive semidefinite precisely when $x^{\mathrm{T}}Ax$ has nonnegative p- and q-spectral components for every $x\in {\mathcal{O}}_{\phi }^n$. If A is spectrally positive definite, then there exists an invertible $L\in M_n\left({\mathcal{O}}_{\phi }\right)$ such that

$$A=L{L}^{\mathrm{T}}.$$

(8.3)

There also exists a unique spectrally positive definite matrix B satisfying

$$B^2=A.$$

(8.4)

More explicitly,

$$L=L_p{e}_p+L_q{e}_q,B=A_p^{1/2}{e}_p+A_q^{1/2}{e}_q,$$

(8.5)

where $L_p,L_q$ are ordinary real Cholesky factors of $A_p,A_q$.

Proof. 

For $x=x_p{e}_p+x_q{e}_q$, the quadratic form splits as

$$x^{\mathrm{T}}Ax=\left(x_p^{\mathrm{T}}{A}_p{x}_p\right)e_p+\left(x_q^{\mathrm{T}}{A}_q{x}_q\right)e_q.$$

If $A_p$ and $A_q$ are positive semidefinite, then both real quadratic forms are nonnegative for all $x_p$ and $x_q$. Conversely, by choosing vectors with arbitrary $x_p$ and $x_q=0$, and then arbitrary $x_q$ and $x_p=0$, the assumed nonnegativity of the two spectral components gives $A_p⪰0$ and $A_q⪰0$.

If A is spectrally positive definite, then $A_p$ and $A_q$ are positive definite real symmetric matrices. Choose ordinary Cholesky factors $A_p=L_p{L}_p^{\mathrm{T}}$ and $A_q=L_q{L}_q^{\mathrm{T}}$ with $L_p,L_q$ invertible. With $L=L_p{e}_p+L_q{e}_q$, Theorem 3.1 gives

$$L{L}^{\mathrm{T}}=L_p{L}_p^{\mathrm{T}}{e}_p+L_q{L}_q^{\mathrm{T}}{e}_q=A.$$

For the square root, let $A_p^{1/2}$ and $A_q^{1/2}$ be the unique real positive definite square roots. Then $B=A_p^{1/2}{e}_p+A_q^{1/2}{e}_q$ is spectrally positive definite and satisfies $B^2=A$. If C is another spectrally positive definite square root, then $C_p^2=A_p$ and $C_q^2=A_q$; real uniqueness gives $C_p=A_p^{1/2}$ and $C_q=A_q^{1/2}$, hence $C=B$. □

Equivalently, the spectrally positive cone is the product cone

$${\mathbb{S}}_n^{+}\left({\mathcal{O}}_{\phi }\right)={\mathbb{S}}_n^{+}\left(\mathbb{R}\right)e_p\oplus {\mathbb{S}}_n^{+}\left(\mathbb{R}\right)e_q,$$

(8.6)

and the Loewner order on symmetric OPNs matrices is the product of the two real Loewner orders. Thus positivity is not an additional order imposed on normalized coordinates; it is the intrinsic cone transported from the two real quadratic-form channels.

9. Matrix Functions, Matrix Equations, and Gauge Equivalence

The final structural applications are functional calculus and matrix equations. These results do not introduce new algebraic mechanisms; they show that standard analytic constructions commute with the same two-channel splitting. Thus exponential flows, Sylvester equations, and Lyapunov equations over OPNs inherit their solvability criteria from the two real channels; the underlying real theory is standard in matrix analysis and linear systems [4,15,16,17].

Theorem 9.1

(Matrix functional calculus and linear matrix equations). Let f be defined on the spectra of $A_p$ and $A_q$ by a convergent power series or by the usual real matrix functional calculus. Define

$$f\left(A\right)=f\left(A_p\right)e_p+f\left(A_q\right)e_q.$$

(9.1)

In particular,

$$exp\left(A\right)=exp\left(A_p\right)e_p+exp\left(A_q\right)e_q,$$

(9.2)

and the initial value problem

$${\displaystyle \frac{dX}{dt}}=AX,X\left(0\right)=X_0$$

(9.3)

has the unique solution

$$X\left(t\right)=exp\left(tA\right)X_0.$$

(9.4)

For $A\in M_m\left({\mathcal{O}}_{\phi }\right)$, $B\in M_n\left({\mathcal{O}}_{\phi }\right)$, and $C\in M_{m,n}\left({\mathcal{O}}_{\phi }\right)$, the Sylvester equation

$$AX-XB=C$$

(9.5)

has a unique solution $X\in M_{m,n}\left({\mathcal{O}}_{\phi }\right)$ for every C precisely when

$$Spec\left(A_p\right)\cap Spec\left(B_p\right)=⌀,Spec\left(A_q\right)\cap Spec\left(B_q\right)=⌀.$$

(9.6)

Let $A,Q\in M_n\left({\mathcal{O}}_{\phi }\right)$ with Q symmetric and spectrally positive definite. The Lyapunov equation

$$A^{\mathrm{T}}X+XA=-Q$$

(9.7)

has a unique spectrally positive definite symmetric solution X precisely when both real matrices $A_p$ and $A_q$ are Hurwitz.

Proof. 

Formula (9.1) is the direct matrix version of the scalar functional calculus. For power series it follows term by term from (3.3); for the usual real matrix functional calculus it is a definition transported through the algebra isomorphism $M_n\left({\mathcal{O}}_{\phi }\right)\cong M_n\left(\mathbb{R}\right)\times M_n\left(\mathbb{R}\right)$. The exponential formula is the special case $f\left(z\right)=e^z$. Differentiating $exp\left(tA\right)X_0$ gives $Aexp\left(tA\right)X_0$, and the value at $t=0$ is $X_0$. Uniqueness follows because (9.3) splits into the two real initial value problems $d{X}_p/dt=A_p{X}_p$ and $d{X}_q/dt=A_q{X}_q$, where ordinary uniqueness applies.

Splitting (9.5) gives the two ordinary real Sylvester equations

$$A_p{X}_p-X_p{B}_p=C_p,A_q{X}_q-X_q{B}_q=C_q.$$

The OPNs equation has a unique solution for every C precisely when each real equation has a unique solution for every corresponding right-hand side. The classical Sylvester criterion gives exactly the two spectral disjointness conditions in (9.6).

Similarly, (9.7) splits into

$$A_p^{\mathrm{T}}{X}_p+X_p{A}_p=-Q_p,A_q^{\mathrm{T}}{X}_q+X_q{A}_q=-Q_q.$$

Since Q is spectrally positive definite, $Q_p$ and $Q_q$ are positive definite. The classical real Lyapunov theorem gives a unique positive definite solution in each channel precisely when $A_p$ and $A_q$ are Hurwitz. Recombining the two solutions gives $X=X_p{e}_p+X_q{e}_q$, and spectral positive definiteness is also channelwise. □

The normalization function is not unique. The preceding results therefore have to be separated from any accidental coordinate choice. The following gauge equivalence states precisely that changing the admissible normalization changes only the normalized presentation of the same spectral algebra. All invariants defined through the p- and q-channels are preserved.

Theorem 9.2

(Gauge equivalence). Let φ and ψ be admissible normalization functions. Define

$$G_{\phi \to \psi }=S_{\psi }^{-1}\circ S_{\phi }:{\mathcal{O}}_{\phi }⟶{\mathcal{O}}_{\psi }.$$

(9.8)

Then $G_{\phi \to \psi }$ is an isomorphism of real algebras. Entrywise application gives algebra isomorphisms

$$M_n\left({\mathcal{O}}_{\phi }\right)\cong M_n\left({\mathcal{O}}_{\psi }\right)$$

(9.9)

that preserve scalar operations, primitive idempotents, finite module decompositions, rank pairs, pivot profiles, determinantal ideal chains, Smith invariant factors, determinants, characteristic polynomials, regular spectral rectangles, similarity classes, double inertias, positivity cones, matrix functions, and matrix equations developed above.

Proof. 

For $\alpha ,\beta \in {\mathcal{O}}_{\phi }$, the transport identities give

$$\begin{array}{cc}\hfill S_{\psi }\left(G_{\phi \to \psi }(\alpha +\beta )\right)& =S_{\phi }(\alpha +\beta )\hfill \\ \hfill & =S_{\phi }\left(\alpha \right)+S_{\phi }\left(\beta \right)\hfill \\ \hfill & =S_{\psi }\left(G_{\phi \to \psi }\left(\alpha \right)\right)+S_{\psi }\left(G_{\phi \to \psi }\left(\beta \right)\right)\hfill \\ \hfill & =S_{\psi }(G_{\phi \to \psi }\left(\alpha \right)+G_{\phi \to \psi }\left(\beta \right)).\hfill \end{array}$$

Injectivity of $S_{\psi }$ proves preservation of addition. Scalar multiplication and multiplication are proved by the same calculation, with the corresponding transport identity in place of addition. Hence $G_{\phi \to \psi }$ is a real-algebra isomorphism.

For matrices, apply $G_{\phi \to \psi }$ entrywise. Since the spectral components are unchanged by construction, every invariant defined through $A_p$ and $A_q$ is preserved. This includes rank pairs, pivot profiles, determinantal ideal chains, Smith invariant factors, determinant components, characteristic polynomial components, spectral rectangles, similarity classes, double inertias, positivity cones, and the four-pivot normal form. □

10. Conclusion

The paper develops OPNs matrix theory from a single structural source: the spectral algebra isomorphism

$${\mathcal{O}}_{\phi }\cong \mathbb{R}{e}_p\oplus \mathbb{R}{e}_q.$$

The scalar theory is not merely a collection of closed formulas on normalized coordinates. It is a reduced semisimple Artinian real algebra with two primitive central idempotents. The normalized pair $(\mu ,\nu )$ supplies a coordinate presentation, while the spectral pair $(p,q)$ supplies the algebraic substance.

This split algebra forces the matrix theory to have two simultaneous real channels and therefore four local pivot types:

$$\tilde{1},e_p,e_q,\tilde{0}.$$

The four-pivot normal form is not a cosmetic rewriting of ordinary rank reduction. The four local pivots are inequivalent, the rank pair is a minimal complete rank-type invariant, and the determinantal ideal chain recovers the rank pair intrinsically. This gives an internal explanation for why a single scalar rank cannot classify OPNs matrices.

The polynomial theory supplies the higher-level source of the constant-matrix normal form. Although ${\mathcal{O}}_{\phi }\left[t\right]$ is not a PID in the integral-domain sense, its idempotent splitting into two copies of $\mathbb{R}\left[t\right]$ gives an idempotent Smith normal form. The same mechanism classifies finitely presented ${\mathcal{O}}_{\phi }\left[t\right]$-modules by paired real PID invariant data. At the non-polynomial level, finitely generated ${\mathcal{O}}_{\phi }$-modules are classified by their two real dimensions, and the canonical summands ${\mathcal{O}}_{\phi }$, ${\mathcal{O}}_{\phi }{e}_p$, and ${\mathcal{O}}_{\phi }{e}_q$ explain the kernel, image, and cokernel formulas attached to the four-pivot form.

For square matrices, determinant theory, adjugates, Cayley–Hamilton, minimal polynomials, invariant factors, regular spectra, similarity, centralizers, symmetric congruence, inertia, positivity, matrix functions, and matrix equations all follow from the same two-channel structure. The final gauge theorem shows that changing the admissible normalization changes only the normalized coordinate presentation, not the intrinsic algebra, module, Smith, similarity, or inertia invariants.

Thus the structural content of OPNs matrix theory is the split semisimple algebra and its irreducible four-pivot linear algebra. The results of the paper are invariant under admissible normalization, complete under row-column equivalence and similarity, and intrinsic at the level of modules and determinantal ideals.