Symmetrizing matrices.

Abstract

A symmetrizing matrix of an arbitrary n-square matrix M is defined as an n-square symmetric matrix B such that BM = M'B. Elementary properties of symmetrizing matrices are established, and an interpretation of a symmetrizing matrix B of M is given with B as the representation of a scalar-product, not necessarily positive definite, with respect to which the arbitrary matrix M, symmetrized by B, represents a self-adjoint operator. Some basic concepts of linear algebra are discussed, leading to a complete derivation of the Jordan canonical form theorem. By considering an arbitrary square matrix in its Jordan canonical form, a complete solution of the symmetrization problem is given, arriving at the results of M. Marcus and N. A. Khan [Pacific J. Math., 10 (1960) 1337-1346]. For the class of nonderogatory matrices it is shown that the symmetrizing matrices are congruent to direct sums of persymmetric matrices. Examples are given of symmetrizing matrices for a companion matrix. References are provided for the construction of symmetrizing matrices of arbitrary matrices.