论文标题
关于1参数矩阵流和静态矩阵的块分解性
On the Block-Decomposability of 1-Parameter Matrix Flows and Static Matrices
论文作者
论文摘要
对于一般复杂或真实的1参数矩阵流量$ a（t）_ {n，n} $，以及时间不变的静态矩阵$ a \ in \ cc_ {n，n} $，本文都认为通过一个常数矩阵流和单个矩阵通过一个常数矩阵$ c_} $ c_} $ c_ { ^{ - 1} \ cdot \ text {diag}（a_1（t），...，a_ \ ell（t））\ cdot c $或$ a = c ^{ - 1} \ cdot \ cdot \ cdot \ text {diag}（a_1，a_1 ,， ... > 1 $如果可能。我们提出的算法背后的理论是基本的，并且使用一个相关流矩阵$ b（t_a）$计算的matlab {\ tt eig}的不变子空间的概念来找到所有流量矩阵$ b（t_b）$。该方法适用于所有时间变化的矩阵流，无论它们在$ t $中都可以区分，连续或不连续，并且对于所有固定的输入矩阵$ a $ a $;以及所有类型的方形矩阵流或固定的入口矩阵，例如Hermitean，真实对称，正常或一般的复杂或真实流量$ a（t）$或静态矩阵$ a $，具有或没有Jordan块结构，并且具有或没有重复的eigenvalues。我们的预期目的是发现对角线块可分解的流动，因为它们起源于传感器驱动的输出，以解决时变矩阵问题，从而有助于通过调整“分裂和征服”方法来减少其数值处理的复杂性，从而减少其对角线亚块。我们的方法还适用于所有结构和类型的标准固定进入矩阵。在此过程中，我们发现并研究了可以在单一相似性下分解为各种$ k $维的块 - 二基因分子形式的K-正常固定进入矩阵类。
For general complex or real 1-parameter matrix flows $A(t)_{n,n}$ and for time-invariant static matrices $A \in \CC_{n,n}$ alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity $C_{n,n}$ as $A(t) = C ^{-1} \cdot \text{ diag}(A_1(t), ..., A_\ell(t)) \cdot C$ or $A = C^{-1}\cdot \text{diag}(A_1,...,A_\ell)\cdot C$ with each diagonal block $A_k(t)$ or $A_k$ square and their number $\ell > 1$ if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the Matlab {\tt 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 very efficiently for all time-varying matrix flows, be they differentiable, continuous or discontinuous in $t$, and for all fixed entry matrices $A$; as well as for all types of square matrix flows or fixed entry matrices such as hermitean, real symmetric, normal or general complex and real flows $A(t)$ or static matrices $A$, with or without Jordan block structures and with or without repeated eigenvalues. Our intended aim is to discover diagonal-block decomposable flows as they originate in sensor driven outputs for time-varying matrix problems and thereby help to reduce the complexities of their numerical treatments through adapting 'divide and conquer' methods for their diagonal sub-blocks. Our method is also applicable to standard fixed entry matrices of all structures and types. 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.
