# Some New Characterizations of a Hermitian Matrix and Their Applications

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## Abstract

A square matrix A over the field of complex numbers is said to be Hermitian if $$A = A^{*}$$, the conjugate transpose of A, while Hermitian matrices are known to be an important class of matrices. In addition to the definition, a Hermitian matrix can be characterized by some other matrix equalities. This fact can be described in the implication form $$f(A, A^{*}) = 0 \Leftrightarrow A = A^{*}$$, where $$f(\cdot )$$ denotes certain ordinary algebraic operation of A and $$A^{*}$$. In this note, we show two special cases of the equivalent facts: $$AA^{*}A = A^{*}AA^{*} \Leftrightarrow A^3 = AA^{*}A \Leftrightarrow A = A^{*}$$ without assuming the invertibility of A through the skillful use of decompositions and determinants of matrices. Several consequences and extensions are presented to a selection of matrix equalities composed of multiple products of A and $$A^{*}$$.

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Correspondence to Yongge Tian.

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Tian, Y. Some New Characterizations of a Hermitian Matrix and Their Applications. Complex Anal. Oper. Theory 18, 2 (2024). https://doi.org/10.1007/s11785-023-01440-x