On the class of Dk-symmetrizable matrices

Abstract

It is known that for every real square matrix A there exists a nonsingular real symmetric matrix S such that SA=A ′S, where A ′ denotes the transpose of A. Using the notion of an M-matrix we derive a criterion for A to satisfy the above equality with a diagonal S of signature k. Such a matrix A will be called D k -symmetrizable and the paper presents some results on this concept. In particular we show that a D k -symmetrizable matrix shares many properties with a real symmetric matrix and that any real matrix A, up to an orthogonal similarity, is D k -symmetrizable for some k.