奇异杨--类矩阵方程的显式解及其数值计算
Explicit Solutions of the Singular Yang--Baxter-like Matrix Equation and Their Numerical Computation
摘要：
当$A$为奇异时，我们得到了求方程$AXA=XAX$无穷多个解的几个显式公式.我们首先分裂成一对线性矩阵方程，然后显示如何投影仪与$A$交换可以用来得到家庭包含一个无限多的解决方案。提出了确定这些投影仪的一些技术，特别是使用Drazin逆，光谱投影仪，矩阵符号函数和特征值的属性。我们还详细研究了如何著名的相似变换，如约旦和舒尔分解可以用来获得新的表示的解决方案。所建议的方法使用有限精度算法的解决方案的计算也是一个问题。在执行过程中出现的困难被确定，并讨论了克服这些困难的想法。数值实验揭示了数值求解该矩阵方程的一些有希望的方法。
英文摘要：
We derive several explicit formulae for finding infinitely many solutions of the equation $AXA=XAX$, when $A$ is singular. We start by splitting the equation into a couple of linear matrix equations and then show how the projectors commuting with $A$ can be used to get families containing an infinite number of solutions. Some techniques for determining those projectors are proposed, which use, in particular, the properties of the Drazin inverse, spectral projectors, the matrix sign function, and eigenvalues. We also investigate in detail how the well-known similarity transformations like Jordan and Schur decompositions can be used to obtain new representations of the solutions. The computation of solutions by the suggested methods using finite precision arithmetic is also a concern. Difficulties arising in their implementation are identified and ideas to overcome them are discussed. Numerical experiments shed some light on the methods that may be promising for solving numerically the said matrix equation.
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