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] denote the leading l × l principal submatrix of A. Let λ (X) ≥ ⋯ ≥ λ (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:{Mathematical expression}The results are extended to equalities involving more than one eigenvalue. As an application, for any 1 ≤ k ≤ n, those A and B that satisfyunderover(∏, i = 1, k) λ (A over(B, ∼)) = underover(∏, i = 1, k) λ (A {ring operator} B),where over(B, ∼) = B or B , the transpose of B, are also determined.