On the structure of the solutions to the matrix equation $G^*JG=J$

TL;DR

This paper analyzes the structure of solutions to the matrix equation G*JG=J, revealing how combining related linear equations simplifies understanding the solution set and its geometric properties, applicable across various algebraic contexts.

Contribution

The paper introduces a novel approach by combining two linear equations to effectively solve and analyze the solution set of G*JG=J, extending to real, complex, and quaternionic cases.

Findings

Explicit solutions for X*J±XJ=0 in real and complex cases

Reduction of problem complexity through combined equations

Application to algebraic and geometric structures like symmetric spaces

Abstract

We study the mathematical structure of the solution set (and its tangent space) to the matrix equation $G^*JG=J$ for a given square matrix $J$. In the language of pure mathematics, this is a Lie group which is the isometry group for a bilinear (or a sesquilinear) form. Generally these groups are described as intersections of a few special groups. The tangent space to $\{G: G^*JG=J \}$ consists of solutions to the linear matrix equation $X^*J+JX=0$. For the complex case, the solution set of this linear equation was computed by De Ter{\'a}n and Dopico. We found that on its own, the equation $X^*J+JX=0$ is hard to solve. By throwing into the mix the complementary linear equation $X^*J-JX=0$, we find that rather than increasing the complexity, we reduce the complexity. Not only is it possible to now solve the original problem, but we can approach the broader algebraic and geometric structure. One implication is that the two equations form an $\mathfrak{h}$ and $\mathfrak{m}$ pair familiar in the study of pseudo-Riemannian symmetric spaces. We explicitly demonstrate the computation of the solutions to the equation $X^*J\pm XJ=0$ for real and complex matrices. However, any real, complex or quaternionic case with an arbitrary involution (e.g., transpose, conjugate transpose, and the various quaternion transposes) can be effectively solved with the same strategy. We provide numerical examples and visualizations.