Working Paper Short Note Version 1 This version is not peer-reviewed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

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