ABSTRACT

The stochastic inverse eigenvalue problem aims to reconstruct a stochastic matrix from its spectrum. Recently, Zhao et al. [A geometric nonlinear conjugate gradient method for stochastic inverse eigenvalue problems. SIAM J Numer Anal. 2016;54(4):2015–2035] proposed a constrained optimization model on the manifold of so-called isospectral matrices and adapted a modified. Polak-Ribière-Polyak conjugate gradient method to the geometry of this manifold. However, not every stochastic matrix is an isospectral one and the model in Zhao et al. is based on the assumption that for each stochastic matrix there exists a (possibly different) isospectral, stochastic matrix with the same spectrum. We are not aware of such a result in the literature, but prove the claim at least for 3 × 3 matrices. In this paper, we suggest to extend the above model by considering matrices which differ from isospectral ones only by multiplication with a block diagonal matrix with 2 × 2 blocks from the special linear group. First, we show that each stochastic matrix can be written in such a form. We prove that our model has a minimizer and show how the Polak–Ribiére–Polyak conjugate gradient method works on the corresponding more general manifold. We demonstrate by numerical examples that the new, more general method performs similarly as those in Zhao et al.

摘要

随机逆特征值问题旨在从其谱中重建随机矩阵。最近，赵等人。[随机逆特征值问题的几何非线性共轭梯度法。暹罗 J 数字肛门。2016;54(4):2015–2035] 在所谓的等谱矩阵的流形上提出了一个约束优化模型，并进行了修改。Polak-Ribière-Polyak 共轭梯度法对此流形的几何结构。然而，并非每个随机矩阵都是等谱矩阵，Zhao 等人的模型。基于这样的假设，即对于每个随机矩阵，存在一个（可能不同的）具有相同光谱的等谱随机矩阵。我们在文献中不知道这样的结果，但至少对 3 × 3 矩阵证明了这一说法。在本文中，我们建议通过考虑与等谱矩阵不同的矩阵来扩展上述模型，仅通过与来自特殊线性群的 2 × 2 块的块对角矩阵相乘。首先，我们证明每个随机矩阵都可以写成这样的形式。我们证明了我们的模型有一个最小值，并展示了 Polak–Ribiére–Polyak 共轭梯度法如何在相应的更一般的流形上工作。我们通过数值示例证明，新的、更通用的方法与 Zhao 等人的方法类似。我们证明了我们的模型有一个最小值，并展示了 Polak–Ribiére–Polyak 共轭梯度法如何在相应的更一般的流形上工作。我们通过数值示例证明，新的、更通用的方法与 Zhao 等人的方法类似。我们证明了我们的模型有一个最小值，并展示了 Polak–Ribiére–Polyak 共轭梯度法如何在相应的更一般的流形上工作。我们通过数值示例证明，新的、更通用的方法与 Zhao 等人的方法类似。