Working Paper Short Note Version 1 This version is not peer-reviewed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

## Subject Areas

Hermitian matrix; matrix decomposition; cancellation property

Views 0