Isospectral matrix flow maintaining staircase structure and total positivity of an initial matrix

Abstract

In this paper we introduce an isospectral matrix flow (Lax flow) that preserves some structures of an initial matrix. This flow is given bydAdt=[Au−Al,A],A(0)=A0, where A is a real n×n matrix (not necessarily symmetric), [A,B]=AB−BA is the matrix commutator (also known as the Lie bracket), Au is the strictly upper triangular part of A and Al is the strictly lower triangular part of A. We prove that if the initial matrix A0 is staircase, so is A(t). Moreover, we prove that this flow preserves the certain positivity properties of A0. Also we prove that if the initial matrix A0 is totally positive or totally nonnegative with non-zero codiagonal elements and distinct eigenvalues, then the solution A(t) converges to a diagonal matrix while preserving the spectrum of A0. Some simulations are provided to confirm the convergence properties.