On the eigenvalues of a matrix with prescribed singular values

Abstract

The singular values of a matrix A (with complex elements) are the non-negative square roots of the eigenvalues of A *A, where A * is the conjugate transpose of A. Let us call a pair of n-tuples (X1, * * , An), (at, * , aC.) allowable if there exists an nth order matrix with eigenvalues '1, * * *, Xn and singular values a,, * * *, an. (Weyl [1 ] has proved that if (X1, * * , Xn), (a,, * * * , an) is allowable and if