Short Note
Version 1
This version is not peer-reviewed
Two Removal and Cancellation Laws Associated with a Complex Matrix and Its Conjugate Transpose
Version 1
: Received: 3 December 2020 / Approved: 4 December 2020 / Online: 4 December 2020 (11:47:55 CET)
How to cite: Tian, Y. Two Removal and Cancellation Laws Associated with a Complex Matrix and Its Conjugate Transpose. Preprints 2020, 2020120103 Tian, Y. Two Removal and Cancellation Laws Associated with a Complex Matrix and Its Conjugate Transpose. Preprints 2020, 2020120103
Abstract
A complex square matrix $A$ is said to be Hermitian if $A= A^{\ast}$, the conjugate transpose of $A$. The topic of the present note is concerned with the characterization of Hermitian matrix. In this note, the we show that each of the two triple matrix product equalities $AA^{\ast}A = A^{\ast}AA^{\ast}$ and $A^3 = AA^{\ast}A$ implies that $A$ is Hermitian by means of decompositions and determinants of matrices, which are named the two-sided removal and cancellation laws associated with Hermitian matrix, respectively. We also present several general removal and cancellation laws as the extensions of the preceding two facts about Hermitian matrix.
Keywords
Hermitian matrix; matrix decomposition; cancellation property
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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