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

Let A and B be n ×n positive semidefinite matrices and 0 < α < β. Let A ∘ B denote the Hadamard product of A and B, and [A]l denote the leading l × l principal submatrix of A. Let λ1(X) ⩾ ⋯ ⩾ 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:λi1/α(AαBα)=λi1/β(AβBβ),i=1,n;λi1/α([Aα]l)=λi1/β([Aβ]l),i=1,…,l;λi1/α(Aα∘Bα)=λi1/β(Aβ∘Bβ),i=1,…,n.The results are extended to equalities involving more than one eigenvalue. As an application, for any 1 ⩽ k ⩽ n, those A and B that satisfy∏i=1kλn-i+1(AB∼)=∏i=1kλn-i+1(A∘B),where B∼=B or BT, the transpose of B, are also determined.