On matrix trace inequalities and related topics for products of Hermitian matrices
An alternate proof of a simple matrix trace inequality of \textit{R. Bellman} [ISNM 47, 89-90 (1980; Zbl 0433.15011)] and \textit{Y. Yang} [J. Math. Anal. Appl. 133, No. 2, 573-574 (1988; Zbl 0651.15018)] is put forward, namely: For positive semidefinite matrices \(A,B\) of the same order, we have \(0 \leq \text{tr} (AB) \leq \text{tr} (A) \text{tr} (B)\). Besides, two theorems on the eigenvalues of two matrices \(A,B\) are proved, namely: (1) If \(A\) and \(B\) are both positive definite, then the eigenvalues of \(AB\) are positive; (2) If \(A,B\) are both Hermitian and of the same order, then the trace of \(AB\) is real.
scientific article; zbMATH DE number 720414