On generalizing trace minimization principles
On generalizing trace minimization principles
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关于推广踪迹最小化原则
DOI：
10.1016/j.laa.2022.10.012
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发表时间：
2023
影响因子：
1.1
通讯作者：
Ren-Cang Li
中科院分区：
文献类型：
--
作者：
Xin Liang;Li Wang;Lei-Hong Zhang;Ren-Cang Li
各种迹最小化原理与一般标准特征值和广义特征值问题以及重要的应用特征值问题的数值计算相互作用，包括电子结构计算的线性响应特征值问题和正定矩阵的辛特征值问题，这些问题在经典哈密顿动力学、量子力学和量子信息等中发挥着重要作用。在本文中，Ky Fan 的迹最小化原理沿着 Stiefel 流形上 X 中的 Brockett 成本函数 tr (D X H A X) 的线进行了扩展，其中适当大小的 D 是正定的。具体来说，我们研究 inf X tr (D X H A X) ，服从 X H B X= I k （k× k 单位矩阵）或 X H B X= J k，其中 J k= diag (±1)。我们建立下确界有限的条件，当它有限时，根据矩阵笔 A− λ B 的特征值和特征向量获得解析解，其中 B 可能是不定的，也可能是奇异的，D 也可能是不定的。
Various trace minimization principles have interplayed with numerical computations for the standard eigenvalue and generalized eigenvalue problems in general, as well as important applied eigenvalue problems including the linear response eigenvalue problem from electronic structure calculation and the symplectic eigenvalue problem of positive definite matrices that play important roles in classical Hamiltonian dynamics, quantum mechanics, and quantum information, among others. In this paper, Ky Fan's trace minimization principle is extended along the line of the Brockett cost function tr (D X H A X) in X on the Stiefel manifold, where D of an apt size is positive definite. Specifically, we investigate inf X tr (D X H A X) subject to X H B X= I k (the k× k identity matrix) or X H B X= J k, where J k= diag (±1). We establish conditions under which the infimum is finite and when it is finite, analytic solutions are obtained in terms of the eigenvalues and eigenvectors of the matrix pencil A− λ B, where B is possibly indefinite and possibly singular, and D is also possibly indefinite.