On the matrix equation $XA + AX^T = 0$

TL;DR

This paper provides an explicit solution to a complex matrix equation related to Lie algebras, advancing understanding where only partial solutions and solution space dimensions were previously known.

Contribution

It offers a new explicit solution for a difficult case of the matrix equation, improving upon prior partial results and algorithms for basis computation.

Findings

Explicit solution for a challenging case of the matrix equation

Determined the dimension of the solution space

Developed an algorithm to find a basis of solutions

Abstract

The matrix equation $XA + AX^T = 0$, which has relevance to the study of Lie algebras, was recently studied by De Teran and Dopico. They reduced the study of this equation to several special cases and produced explicit solutions in most instances. In this note we obtain an explicit solution in one of the difficult cases, for which only the dimension of the solution space and an algorithm to find a basis of this space were known previously.