关于酉结构保持变换下的正规矩阵与结构化矩阵

Sampling the eigenvalues of random orthogonal and unitary matrices

We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such matrices, and then computes their eigenvalues with a tailored core-chasing algorithm. This approach requires a number of floating-point operations that is quadratic in the order of the matrix being sampled, and can be adapted to other matrix groups. In particular, we explain how it can be used to sample the Haar measure over the special orthogonal and unitar…

线性结构矩阵的结构映射问题

Structured mapping problems for linearly structured matrices

研究线性结构矩阵的映射特性，确定满足AX=B的矩阵A及映射X到B的矩阵，并分析最小范数解，用于探讨近似特征对和不变子空间的结构回溯误差及结构伪谱。

Fast QR iterations for unitary plus low rank matrices

Some fast algorithms for computing the eigenvalues of a block companion matrix $A = U + XY^H$, where $U\in \mathbb C^{n\times n}$ is unitary block circulant and $X, Y \in\mathbb{C}^{n \times k}$, have recently appeared in the literature. Most of these algorithms rely on the decomposition of $A$ as product of scalar companion matrices which turns into a factored representation of the Hessenberg reduction of $A$. In this paper we generalize the approach to encompass Hessenberg matrices of the form $A=U + XY^H$ where $U$ is a general unitary matrix. A remarkable case is $U$ unitary diagonal which…

将单位ary秩结构化矩阵压缩为CMV-like形状及其在多项式根寻找中的应用

Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding

本文提出了一种将单位ary矩阵压缩为CMV-like形状的Lanczos型算法，并讨论其在多项式根寻找中的应用。

Efficient Reduction of Compressed Unitary plus Low-rank Matrices to Hessenberg form

We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix $A=G+U V^H$, where $G\in \mathbb C^{n\times n}$ is a unitary matrix represented in some compressed format using $O(nk)$ parameters and $U$ and $V$ are $n\times k$ matrices with $k< n$. At the core of these methods is a certain structured decomposition, referred to as a LFR decomposition, of $A$ as product of three possibly perturbed unitary $k$ Hessenberg matrices of size $n$. It is shown that in most interesting cases an initial LFR decomposition of $A$ can be computed very cheaply. Then…

非退化矩阵的酉相似性

Unitary Similarity of Nonderogatory Matrices

针对非退化矩阵的酉相似性验证问题，提出一种基于几何方法的稳定且高效的数值解法。