论文标题
结构化有理矩阵的结构性强线性化
Structured strong linearizations of structured rational matrices
论文作者
论文摘要
在许多应用中，在许多应用中都出现了结构化的有理矩阵，例如对称，偏斜，哈密顿，偏斜 - 哈米尔顿，赫尔米尔顿和帕拉 - 温米特理性矩阵。最近引入了理性矩阵的线性化，用于计算电线杆，特征值，特征向量，最小碱基和最小矩阵指数。对于结构化的有理矩阵，希望构建具有结构的线性化，以保留有理矩阵的特征值和极点中的对称性。为了构建结构化有理矩阵结构的线性化，我们提出了一个类似Fiedler的铅笔的家族，并表明类似Fiedler的铅笔家族是结构性结构化理性矩阵的结构强度线性化的丰富来源。我们构建对称，偏斜的对称，哈密顿式，偏斜 - 哈米尔顿，赫尔米尔顿，偏斜 - 热，para-hermitian和para-skew-skew-skew-skew-skew-skew-hermitian强矩阵$ g（λ）$时，$ g（λ）$具有相同的结构。此外，当$ g（λ）$是真实的和对称的时，我们表明，$ g（λ）$的真实对称线性化的传输函数保留了$ g（λ）的库奇·马斯洛夫指数。无操作。
Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. With a view to constructing structure-preserving linearizations of structured rational matrices, we propose a family of Fiedler-like pencils and show that the family of Fiedler-like pencils is a rich source of structure-preserving strong linearizations of structured rational matrices. We construct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrix $G(λ)$ when $G(λ)$ has the same structure. Further, when $G(λ)$ is real and symmetric, we show that the transfer functions of real symmetric linearizations of $G(λ)$ preserve the Cauchy-Maslov index of $G(λ).$ We describe the recovery of eigenvectors, minimal bases and minimal indices of $G(λ)$ from those of the linearizations of $G(λ)$ and show that the recovery is operation-free.
