Eigenvalue equalities for ordinary and Hadamard products of powers of positive semidefinite matrices Original Research Article

Let A and B be n ×n positive semidefinite matrices and 0 < α < β. Let A ring operator B denote the Hadamard product of A and B, and [A]l denote the leading l × l principal submatrix of A. Let λ1(X) greater-or-equal, slanted cdots, three dots, centered greater-or-equal, slanted λn(X) denote the eigenvalues of an n × n matrix X ordered when they are all real. In this paper, those matrices that satisfy any of the following equalities are determined:imageThe results are extended to equalities involving more than one eigenvalue. As an application, for any 1 less-than-or-equals, slant k less-than-or-equals, slant n, those A and B that satisfyimagewhere image or BT, the transpose of B, are also determined.