Eigenvalue equalities for ordinary and Hadamard products of powers of positive semidefinite matrices (Q875033)

Let \(A\) and \(B\) be \(n\times n\) positive semidefinite matrices and \(0<\alpha <\beta.\) Let \(A\circ B\) denote the Hadamard product of \(A\) and \(B,\) and \([A]_l\) denote the leading \(l\times l\) principle submatrix of \(A.\) Let \(\lambda_1(X)\geq \cdots \geq \lambda_n(X)\) denote the eigenvalues of an \(n\times n\) matrix \(X\) ordered when they are all real. In this paper, those matrices that satisfy any of the equalities \[ \lambda _i^{1/\alpha }(A^\alpha B^\alpha)=\lambda _i^{1/\beta }(A^\beta B^\beta),\quad i=1,n \] \[ \lambda _i^{1/\alpha }([A^\alpha ]_l)=\lambda _i^{1/\beta }([A^\beta ]_l ),\quad i=1,\dots ,l \] \[ \lambda _i^{1/\alpha }(A^\alpha \circ B^\alpha)=\lambda _i^{1/\beta }(A^\beta \circ B^\beta),~~i=1,\dots ,n \] are determined. Several related results are discussed.