On the similarity transformation between a matrix and its transpose

Abstract

It was observed by one of the authors that a matrix transforming a companion matrix into its transpose is symmetric. The following two questions arise: I. Does there exist for every square matrix with coefficients in a field a non-singular symmetric matrix transforming it into its transpose ? II. Under which conditions is every matrix transforming a square matrix into its transpose symmetric?. © 1959 by Pacific Journal of Mathematics.