Let A∈C;,set eigenvalues of matrix A with |λ;(A)|≥|λ;(A)|≥…≥|λ;(A)|,write A≥0 if A is a positive semidefinite Hermitian matrix, and denote∧;(A)=diag (λ;(A),…,λ;(A)),∧;(n-k).(A)=diag (λ;(A),…,λ;(A))for any k=1, 2,...,n if A≥0. Denote all n order unitary matrices by U;.Problem of equalities to hold in eigenvalue inequalities for products of matrices was