On the matrix equation $XA+AX^T =0$, II: Type 0-I interactions

TL;DR

This paper provides an explicit, closed-form solution to the matrix equation $XA + AX^T = 0$ specifically for the challenging Type 0-I interaction case, advancing understanding of congruence orbit dimensions.

Contribution

It introduces a complete explicit solution for the Type 0-I interaction case of the matrix equation, filling a gap in prior research.

Findings

Explicit solution for Type 0-I case obtained

Advances understanding of congruence orbit dimensions

Completes analysis of special cases of the matrix equation

Abstract

The matrix equation $XA + AX^T = 0$ was recently introduced by De Ter\'an and Dopico to study the dimension of congruence orbits. They reduced the study of this equation to a number of special cases, several of which have not been explicitly solved. In this note we obtain an explicit, closed-form solution in the difficult Type 0-I interaction case.