$T$ -回文束的不变子空间与代数$T$ -Riccati方程
Invariant subspaces of $T$-palindromic pencils and algebraic $T$-Riccati equations
摘要：
利用求解代数$\top$ -Riccati方程和计算$\top$ -回文矩阵束的某些收缩子空间之间的联系，我们得到了这两个问题的理论和计算结果.在理论上，我们引入了避免$\top$ -回文矩阵束模1特征值存在的条件和$\top$ -Riccati方程解存在的条件.在计算上，我们改进了回文QZ算法与一个新的排序过程，并引入新的算法来计算紧缩子空间的$\top$ -回文铅笔，基于二次化的铅笔或上的整表示的正交投影寻求紧缩子空间。
英文摘要：
By exploiting the connection between solving algebraic $\top$-Riccati equations and computing certain deflating subspaces of $\top$-palindromic matrix pencils, we obtain theoretical and computational results on both problems. Theoretically, we introduce conditions to avoid the presence of modulus-one eigenvalues in a $\top$-palindromic matrix pencil and conditions for the existence of solutions of a $\top$-Riccati equation. Computationally, we improve the palindromic QZ algorithm with a new ordering procedure and introduce new algorithms for computing a deflating subspace of the $\top$-palindromic pencil, based on quadraticizations of the pencil or on an integral representation of the orthogonal projector on the sought deflating subspace.
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