On the structure of the solutions to the matrix equation G⁎JG = J
On the structure of the solutions to the matrix equation G⁎JG = J
复制标题
矩阵方程G−JG−=−J的解的结构
DOI：
10.1016/j.laa.2022.10.007
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发表时间：
2023
影响因子：
1.1
通讯作者：
Jeong, Sungwoo
中科院分区：
文献类型：
--
作者：
Edelman, Alan;Jeong, Sungwoo
我们研究了矩阵方程G = J的解集（及其切线空间）的数学结构，对于给定的方阵J，这是一个李群，它是双线性（或半双线性）形式的等距群。一般来说，这些群体被描述为一些特殊群体的交叉点。{G：G <$J G= J}的切空间由线性矩阵方程X <$J+ J X= 0的解组成。对于复杂的情况，这个线性方程的解集由De Terán和Dopico计算。我们发现，方程X <$J+ J X= 0本身很难求解。通过将互补线性方程X <$J− J X= 0放入混合中，我们发现两个解集的直和更容易计算线性空间。因此，我们从投影映射得到两个解集。现在不仅有可能解决原来的问题，而且我们可以接近更广泛的代数和几何结构。一个含义是，这两个方程形成了一个在伪黎曼对称空间研究中常见的h和m对。我们明确地证明了方程X <$J± XJ = 0的真实的矩阵和复矩阵的解的计算。然而，真实的，复杂的或四元数的情况下，任意的对合（例如，转置，共轭转置，和各种四元数转置）可以有效地解决了相同的策略。我们提供数值例子和可视化。
We study the mathematical structure of the solution set (and its tangent space) to the matrix equation G⁎ J G= 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⁎ J G= J} consists of solutions to the linear matrix equation X⁎ J+ J X= 0. For the complex case, the solution set of this linear equation was computed by De Terán and Dopico. We found that on its own, the equation X⁎ J+ J X= 0 is hard to solve. By throwing into the mix the complementary linear equation X⁎ J− J X= 0, we find that the direct sum of the two solution sets is an easier to compute linear space. Thus, we obtain the two solution sets from projection maps. 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 h and m pair familiar in the study of pseudo-Riemannian symmetric spaces. We explicitly demonstrate the computation of the solutions to the equation X⁎ J±X J= 0 for real and complex matrices. However, real, complex or quaternionic case with an arbitrary involution (eg, transpose, conjugate transpose, and the various quaternion transposes) can be effectively solved with the same strategy. We provide numerical examples and visualizations.