How a Unitoid Matrix Loses Its Unitoidness?

IF 0.4 Q4 MATHEMATICS, APPLIED

Numerical Analysis and Applications Pub Date : 2024-09-13 DOI:10.1134/s1995423924030029

Kh. D. Ikramov, A. M. Nazari

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Abstract

A unitoid is a square matrix that can be brought to diagonal form by a congruence transformation. Among different diagonal forms of a unitoid \(A\), there is only one, up to the order adopted for the principal diagonal, whose nonzero diagonal entries all have the modulus 1. It is called the congruence canonical form of \(A\), while the arguments of the nonzero diagonal entries are called the canonical angles of \(A\). If \(A\) is nonsingular then its canonical angles are closely related to the arguments of the eigenvalues of the matrix \(A^{-*}A\), called the cosquare of \(A\). Although the definition of a unitoid reminds the notion of a diagonalizable matrix in the similarity theory, the analogy between these two matrix classes is misleading. We show that the Jordan block \(J_{n}(1)\), which is regarded as an antipode of diagonalizability in the similarity theory, is a unitoid. Moreover, its cosquare \(C_{n}(1)\) has \(n\) distinct unimodular eigenvalues. Then we immerse \(J_{n}(1)\) in the family of the Jordan blocks \(J_{n}(\lambda)\), where \(\lambda\) is varying in the range \((0,2]\). At some point to the left of 1, \(J_{n}(\lambda)\) is not a unitoid any longer. We discuss this moment in detail in order to comprehend how it can happen. Similar moments with even smaller \(\lambda\) are discussed, and certain remarkable facts about the eigenvalues of cosquares and their condition numbers are pointed out.

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类空矩阵如何失去类空性？

AbstractA unitoid 是一个可以通过全等变换转换成对角线形式的正方形矩阵。在单位体 \(A\) 的不同对角线形式中，只有一种在主对角线所采用的阶以内，其非零对角线项的模数都是 1。它被称为 \(A\) 的全等规范形式，而非零对角线项的参数被称为 \(A\) 的规范角。如果 \(A\) 是非正交的，那么它的正交角就与矩阵 \(A^{-*}A\) 的特征值的参数密切相关，称为 \(A\) 的余弦平方。虽然单元体的定义让人想起相似性理论中可对角矩阵的概念，但这两类矩阵之间的类比是有误导性的。我们将证明，在相似性理论中被视为可对角化矩阵反节点的乔丹块 \(J_{n}(1)\)是一个类单元。此外，其余弦 \(C_{n}(1)\) 有 \(n\) 个不同的单调特征值。然后我们把\(J_{n}(1))浸入约旦块族\(J_{n}(\lambda)\)中，其中\(\lambda\)在\((0,2]\)范围内变化。在 1 左边的某一点上， \(J_{n}(\lambda)\) 不再是单位体了。我们将详细讨论这个时刻，以理解它是如何发生的。我们还讨论了具有更小的\(\lambda\)的类似时刻，并指出了关于余弦特征值及其条件数的某些显著事实。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Numerical Analysis and Applications MATHEMATICS, APPLIED-

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1.00

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期刊介绍： Numerical Analysis and Applications is the translation of Russian periodical Sibirskii Zhurnal Vychislitel’noi Matematiki (Siberian Journal of Numerical Mathematics) published by the Siberian Branch of the Russian Academy of Sciences Publishing House since 1998. The aim of this journal is to demonstrate, in concentrated form, to the Russian and International Mathematical Community the latest and most important investigations of Siberian numerical mathematicians in various scientific and engineering fields. The journal deals with the following topics: Theory and practice of computational methods, mathematical physics, and other applied fields; Mathematical models of elasticity theory, hydrodynamics, gas dynamics, and geophysics; Parallelizing of algorithms; Models and methods of bioinformatics.