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.
翻译:我们研究矩阵方程式的数学结构( 及其正切空间) $G ⁇ JG=J$ 的数学结构 。 在纯数学语言中, 这是一个 Lie 组, 是双线( 或正线) 形式的等离度组 。 这些组通常被描述为几个特殊组的交叉点 。 淡度空间为 $G: G ⁇ JG=J =J $, 包括线性矩阵方程式的解决方案 $X ⁇ J+JX=0 美元 。 在复杂的情况下, 这个线性方程式的解决方案由 De Ter@a}n 和 Dopico 计算 。 我们发现, 单方的等值组是双等值的双线性方 $X+J+J= =0 美元。 我们发现, 不仅增加了复杂性, 我们还可以解决任何原始问题, 但我们可以用更宽的平面和几何方程的方程方程结构来解析度 。 一种暗示, 两个等式的内数是 。