Working Paper Short Note Version 1 This version is not peer-reviewed

# Analytical Formulas for Calculating Ranks, Dimensions, Orthogonal Projectors, and Ranges of Matrices Composed of Kronecker Products

Version 1 : Received: 26 August 2020 / Approved: 31 August 2020 / Online: 31 August 2020 (04:57:21 CEST)

How to cite: Tian, Y. Analytical Formulas for Calculating Ranks, Dimensions, Orthogonal Projectors, and Ranges of Matrices Composed of Kronecker Products. Preprints 2020, 2020080698 Tian, Y. Analytical Formulas for Calculating Ranks, Dimensions, Orthogonal Projectors, and Ranges of Matrices Composed of Kronecker Products. Preprints 2020, 2020080698

## Abstract

Kronecker products of matrices have some striking operation properties, one of which is called the mixed-product property $(A \!\otimes\! B)(C \!\otimes\! D) = AC \!\otimes\! BD$. In view of this property, the two-term Kronecker product $A_1 \otimes A_2$ can be rewritten as $A_1 \otimes A_2 = (A_1 \otimes I_{m_2})(I_{m_1} \otimes A_2)$ of dilation forms of $A_1$ and $A_2$, and three-term Kronecker product $A_1 \otimes A_2 \otimes A_3$ can be rewritten as the products $A_1 \otimes A_2 \otimes A_3 = (A_1\otimes I_{m_2} \otimes I_{m_3})(I_{m_1} \otimes A_2 \otimes I_{m_3})(I_{m_1} \otimes I_{m_2} \otimes A_3)$ of the dilation forms of $A_1$, $A_2$, and $A_3$, respectively, where the matrices on the right-hand sides of the two factorizations are commutative. In this note, we approach the commutative Kronecker products on the right-hand sides of the two factorization equalities, and present a variety of new and useful analytical formulas for calculating the ranks, dimensions, orthogonal projectors, and ranges of matrices composed of these Kronecker products.

## Keywords

Kronecker product; rank; dimension; orthogonal projector; range

## Subject

Computer Science and Mathematics, Mathematics