四元数矩阵的酉相合与酉相似

本文讨论了四元数矩阵的酉相合,四元数矩阵的酉相合是复矩阵的复酉相合的自然推广,并且它有许多好的性质.由于四元数矩阵的酉相合与酉相似有着密切联系,本文还讨论了四元数矩阵的酉相似的一些性质.     

善用初等变换展开线性代数理论教学

初等变换是线性代数的基本变换,在线性代数课程中常常被用来计算,例如求解线性方程组、计算方阵的行列式、矩阵的求逆以及更一般的矩阵方程AX=B的求解、计算整数矩阵和域上多项式矩阵的Smith标准形、以及计算对称阵的相合标准形等.本文说明如何灵活利用初等变换,给出线性代数课程中一些重要理论结果的系统而又简洁的证明.     

算子值酉随机矩阵及其渐近自由性

该文在算子值非交换概率空间上引入半标准酉随机矩阵的概念, 证明了它是算子值Haar酉元的矩阵模型,并给出了半标准酉随机矩阵的渐近自由判定定理.     

关于矩阵多项式Jordan标准形刻画

本文利用矩阵运算、矩阵相似关系及矩阵的秩,深化了Jordan矩阵的性质,并在此基础上刻画了矩阵Jordan标准形中Jordan块的个数及阶数,最后讨论了矩阵多项式Jordan标准形,充实了高等代数中Jordan标准形的结果．     

具有k重零特征根的矩阵的性质

本文对具有 k重零特征根的矩阵的一些性质进行了探讨 ,这些性质主要涉及到矩阵的秩、矩阵的有理标准形和约当标准形等方面     

层次分析正互反矩阵的Hadamard乘积分析法

定义了标准形 ～ 型 ,并指出任一正互反矩阵可唯一分解为任一种标准形和一个一致性矩阵的Hadamard乘积 .     

酉对称矩阵的极分解

为了简化大型行(列)酉对称矩阵的极分解,研究了酉对称矩阵的性质,获得了一些新的结果,给出了酉对称矩阵的极分解和广义逆的公式,它们可极大地减少行(列)酉对称矩阵的极分解的计算量与存储量,并且不会丧失数值精度.同时对酉对称矩阵的极分解作了扰动分析.     

初等变换

讨论矩阵计算中初等变换的意义和一些性质,实例展示了初等变换在求逆矩阵、等价标准形、线性空间的过渡矩阵以及二次型的标准形等方面的实际应用．     

关于矩阵相似中变换矩阵的一般形式

本文利用 Jordan标准形理论 ,给出了矩阵相似中的变换矩阵的一般表达形式 .     

关于酉对称矩阵的QR分解及其算法

为了简化大型行(列)酉对称矩阵的QR分解,研究了行(列)酉对称矩阵的性质,获得了一些新的结果,给出了行(列)酉对称矩阵的QR分解的公式和快速算法,它们可极大地减少行(列)酉对称矩阵的QR分解的计算量与存储量,并且不会丧失数值精度.同时推广和丰富了邹红星等(2002)的研究内容,拓宽了实际应用领域的范围.     

Canonical forms for unitary congruence and *congruence

We use methods of the general theory of congruence and *congruence for complex matrices – regularization and cosquares – to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that āA (respectively, A ²) is normal. As special cases of our canonical forms, we obtain – in a coherent and systematic way – known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, λ-projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A ³ is normal, and (b) unitary congruence when AāA is normal, are both unitarily wild, so these classification problems are hopeless.     

Some observations on the Youla form and conjugate-normal matrices

Two square complex matrices A, B are said to be unitarily congruent if there is a unitary matrix U such that A = UBU^T. The Youla form is a canonical form under unitary congruence. We give a simple derivation of this form using coninvariant subspaces. For the special class of conjugate-normal matrices the associated Youla form is discussed.     

On tridiagonal conjugate-normal matrices

Conjugate-normal matrices play the same role in the theory of unitary congruences as conventional normal matrices do with respect to unitary similarities. Naturally, the properties of both matrix classes are fairly similar up to the distinction between the congruence and similarity. However, in certain respects, conjugate-normal matrices differ substantially from normal ones. Our goal in this paper is to indicate one of such distinctions. It is shown that none of the familiar characterizations of normal matrices having the irreducible tridiagonal form has a natural counterpart in the case of conjugate-normal matrices.     

Reducing complex matrices to condensed forms by unitary congruence transformations

We show that any n × n conjugate-normal matrix can be brought by a unitary congruence transformation to block-tridiagonal form with the orders of the consecutive diagonal blocks not exceeding 1, 2, 3, ..., respectively. The proof is constructive; namely, a finite process is described that implements the reduction to the desired form. Sufficient conditions are indicated for the orders of the diagonal blocks to stabilize. In this case, the condensed form is a band matrix.     

Generalized normal matrices

A matrix A ∈ Mn（C） is called generalized normal provided that there is a positive definite Hermite matrix H such that HAH is normal. In this paper, these matrices are investigated and their canonical form, invariants and relative properties in the sense of congruence are obtained.     

对称部分为半正定的方阵

所有ｎ阶具有半正定对称部分的方阵的集合记作ＰＳＤｎ．本文给出了ＰＳＤｎ中方阵在合同下的标准形以及ＰＳＤｎ中两个方阵合同的一个充要条件，并给出了ＰＳＤｎ中一个方阵及其对称部分与斜对称部分的主子式间的一个不等式．     

Classification of squared normal operators on unitary and Euclidean spaces

We give a canonical form for a complex matrix whose square is normal under transformations of unitary similarity as well as a canonical form for a real matrix whose square is normal under transformations of orthogonal similarity. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 225–232, 2007.     

Reachability of condensed forms via unitary congruence transformations

There are several well-known facts about unitary similarity transformations of complex n-by-n matrices: every matrix of order n = 3 can be brought to tridiagonal form by a unitary similarity transformation; if n ≥ 5, then there exist matrices that cannot be brought to tridiagonal form by a unitary similarity transformation; for any fixed set of positions (pattern) S whose cardinality exceeds n(n ? 1)/2, there exists an n-by-n matrix A such that none of the matrices that are unitarily similar to A can have zeros in all of the positions in S. It is shown that analogous facts are valid if unitary similarity transformations are replaced by unitary congruence ones.     

一个被遗漏的Hermite标准形的重要性质及应用

为了使Hermite标准形得到更好的应用,给出了左、右单位矩阵的概念,并指出了矩阵的Hermite行(列)标准形与该矩阵的右(左)单位矩阵的关系,且将其灵活地运用到矩阵广义逆及矩阵方程的讨论中去.