For general complex or real 1-parameter matrix flows A(t)_{n, n} and for static matrices \(A \in \mathbb {C}_{n,n}\) alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity C_{n, n} as A(t) = C^− 1 ⋅ diag(A₁(t),...,A_ℓ(t)) ⋅ C or A = C^− 1 ⋅diag(A₁,...,A_ℓ) ⋅ C with each diagonal block A_k(t) or A_k square and their number ℓ exceeding 1 if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the MATLAB eig computed ‘eigenvectors’ of one associated flow matrix B(t_a) to find the coarsest simultaneous block structure for all flow matrices B(t_b). The method works efficiently for all time-varying matrix flows A(t), be they real or complex, normal, with Jordan structures or repeated eigenvalues, differentiable, continuous, or discontinuous in t, and likewise for all fixed entry matrices A. Our intended aim is to discover unitarily diagonal-block decomposable flows as they originate in real-time from sensor given data for time-varying matrix problems that are unitarily invariant. Then, the complexities of their numerical treatments decrease by adopting ‘divide and conquer’ methods for their diagonal blocks. In the process, we discover and study k-normal fixed entry matrix classes that can be decomposed under unitary similarities into various k-dimensional block-diagonal forms.

对于一般的复数或实数一参数矩阵流A（t）_n，n和静态矩阵\（A \ in \ mathbb {C} _ {n，n} \）而言，本文都考虑了分解矩阵流的方法，全局单矩阵经由一个常数矩阵相似ç _ñ，ñ如甲（吨）= c ^ ^{- 1} ⋅DIAG（阿₁（吨），...，甲_ℓ（吨））⋅ ç或甲= c ^ ^{- 1个}⋅diag （A ₁，...，甲_ℓ）⋅ Ç与每个对角块甲_ķ（吨）或甲_ķ正方形，其数量ℓ超过1，如果这是可能的。我们提出的算法背后的理论是基础知识，它使用不变子空间的概念来为一个相关联的流矩阵B（t _a）进行MATLAB eig计算的``特征向量''，以找到所有流矩阵B（t _b）的最粗糙的同时块结构。该方法对于所有时变矩阵流A（t），无论它们是真实的还是复杂的，正常的，具有约旦结构或重复的特征值，在t中是可微的，连续的或不连续的，并且对于所有固定进入矩阵A都是相同的。我们的预期目标是发现统一的对角块可分解流，因为它们实时源自传感器给定数据的时不变矩阵时变矩阵问题。然后，通过对角线块采用“分而治之”的方法来降低其数字处理的复杂性。在这一过程中，我们发现并研究了可在单一相似性下分解为各种k维块对角线形式的k正态固定项矩阵类别。