On Constructing Matrices with Prescribed Singular Values and Diagonal Elements

Similar to the well known Schur Horn theorem that characterizes the relationship between the diagonal entries and the eigenvalues of a Hermitian matrix the Sing Thompson theorem characterizes the relationship between the diagonal en tries and the singular values of an arbitrary matrix It is noted in this paper that based on the induction principle such a matrix can be constructed numerically by a fast recursive algorithm provided that the given singular values and diagonal elements satisfy the Sing Thompson conditions Introduction It has been observed that the main diagonal en tries and the eigenvalues of any Hermitian matrix enjoy an interesting relationship This relationship is completely characterized by what is now known as the Schur Horn theorem For reference we describe speci cally the theorem in two parts More details and related topics can be found for example in Theorems and Theorem Schur Horn Theorem Given an arbitrary Hermitian matrix H let i R n and a ai R n denote the vectors of eigenvalues and main diagonal entries of H respectively If the entries are arranged in increasing order aj ajn and m mn then