Realizations of interlacing by tree-patterned matrics

Abstract

In this note, we prove the following theorem. Given 2n real numbers x ₁≥β₁≥x,≥≥β₂≥ [cddot] x _n ≥β _n ≥0 and a directed tree T on n+1 vertices with a distinguished vertex which is a source. then there exists an n+1 x n real matrix A that satisfies.

1. The directed graph of A is T (where the first row of A corresponds to the distinguished source vertex);

2. The singular values of A are α₁, …, α _n

3. The singular values of the n × n matrix A ₀ obtained by deleting the top row of A are β₁, …, β _n

We adapt out method to prove a similar result first found by A. L. Duarte for eigenvalue interlacing of Hermitian matrices. Some partial results of this type for invariant factor interlacing of integral matrices are given.