Some New Characterizations of a Hermitian Matrix and Their Applications.

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 ⇔ A = A ∗ , where f (·) denotes certain ordinary algebraic operation of A and A ∗ . In this note, we show two special cases of the equivalent facts: A A ∗ A = A ∗ A A ∗ ⇔ A 3 = A A ∗ A ⇔ 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 ∗ . [ABSTRACT FROM AUTHOR]