对角量子信道纠错码空间的存在性与构造

通过量子信道的Kraus算子,提出了对角量子信道的概念,证明了对角量子信道的一些性质:一个量子信道成为对角量子信道的充要条件是所有对角矩阵都是它的不动点;同一对角量子信道的所有压缩矩阵具有相同的秩;一个对角量子信道不可纠错的充要条件是其压缩矩阵是行满秩的.进而证明了一个对角量子信道在整个空间上可纠错当且仅当其压缩矩阵为1秩阵.最后,利用一个具体例子给出了构造对角量子信道的码空间的一种方法.     

关于矩阵的可对角化

本文利用矩阵秩、矩阵相似、最小多项式及特殊矩阵的特性，讨论了利用矩阵秩判断矩阵可对角化的充要条件及典型的特殊矩阵类对角化问题.     

矩阵可对角化的一个充要条件

本文给出矩阵可对角化(即可与对角矩阵相似)的一个充要条件,并推广了文[1]中的一个结果。首先叙述如下: 引理设A,B都是n阶矩阵,则有秩(AB)≥秩A+秩B-n 证明可见[2],这里从略。定理设A是数域F上的一个n阶矩阵,     

线性代数中特征向量的一个应用

在线性代数中,特征向量在矩阵的对角化过程中起着重要作用.从一个引例出发,证明了:一个矩阵与对角矩阵可交换当且仅当它可以用以特征向量为列向量的两个矩阵表示.做为推论,如果对角矩阵对角线上的相同元素在相邻位置,那么与其可交换的矩阵只能是准对角矩阵.     

四元数自共轭矩阵乘积的相似变换

本文证明了下列结果:(i)四元数矩阵A可写成两个自共轭四元数矩阵的乘积A相似于实矩阵A Hermite相似于A~*.(ii)A可写成一个半正定自共轭四元数矩阵与一个自共轭四元数矩阵的乘积A相似于实对角矩阵或者A～diag(D,I_r(×)J_2(O)),其中D是一个实对角矩阵.本文还给出了体上实矩阵AB与BA相似的一个充要条件.     

一类高阶矩阵差分方程的解及渐近稳定性

讨论了一类高阶矩阵差分方程的解及渐近稳定性问题.利用特征子空间的维数得到了特征方程存在可对角化解的一个充要条件;然后利用特征方程的相异解刻划出该矩阵差分方程的通解,并给出其解渐近稳定的两个充分条件.推广了相关文献的结果.     

有界上三角算子矩阵的亏谱

讨论了希尔伯特空间上有界上三角算子矩阵的亏谱扰动性质,当对角元算子给定时,得到上三角算子矩阵的亏谱恰等于对角元算子的亏谱之并集的充要条件,特别地,给出有界上三角Hamilton型算子矩阵相应问题成立的条件,并辅以实例佐证.     

λ-矩阵的等价和矩阵多项式秩的恒等式

设U(λ)与V(λ)都是m×m阶的λ-矩阵.若U(λ)与V(λ)等价,则对于任意的n阶方阵A,分块矩阵U(A)与V(A)的秩相等.利用此结论刻画了幂零矩阵、零化多项式等.同时,通过考虑两个对角λ-矩阵等价的充要条件,使关于矩阵多项式秩的一些恒等式的讨论有了新的统一的方法.     

关于分块矩阵〔ABCD〕的g-逆的块独立性

本文证明了分块阵M=〔ABCD〕的g-逆有块独立性的充要条件是M适合秩可加性条件,M~+有一特定的表达式的充要条件也是M适合秩可加性条件.本文还给出了包含M的g-逆的子块的不变矩阵以及这些子块的定义方程.     

矩阵的秩的一个定理和线性方程组的同解定理

本文给出了矩阵乘积的秩定理的一个逆形式,并应用它证明了线性方程组的同解定理. 本文中的符号同[1].在[1]中有以下定理: 定理:两个矩阵的乘积的秩不大于每一因子的秩.特别,当有一个因子是可逆矩阵时,乘积的秩等于另一因子的秩.     

Linear preservers of minimal rank

It is proved that a linear transformation on the vector space of upper triangular matrices that maps the set of matrices of minimal rank 1 into itself, and either has the analogous property with respect to matrices of full minimal rank, or is bijective, is a triangular equivalence, or a flip about the south-west north-east diagonal followed by a triangular equivalence. The result can be regarded as an analogue of Marcus–Moyls theorem in the context of triangular matrices.     

Positive matrix semigroups with binary diagonals

We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank in S{\mathcal{S}} satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable sequences.     

The elementary divisor theory over the Hurwitz order of integral quaternions

There exists a diagonal form with certain divisibility conditions for matrices over the Hurwitz order of integral quaternions under unimodular equivalence. The diagonal entries are uniquely determined up to similarity. Given two such diagonal forms, where the diagonal entries are similar by pairs, the matrices prove to be ummodularly equivalent, whenever the rank of the matrices is creater than one.     

ON STABLE PERTURBATIONS OF THE STIFFLY WEIGHTED PSEUDOINVERSE AND WEIGHTED LEAST SQUARES PROBLEM

In this paper we study perturbations of the stiffly weighted pseudoinverse （W^1/2 A）^＋W^1/2 and the related stiffly weighted least squares problem, where both the matrices A and W are given with W positive diagonal and severely stiff. We show that the perturbations to the stiffly weighted pseudoinverse and the related stiffly weighted least squares problem are stable, if and only if the perturbed matrices A = A ＋ δA satisfy several row rank preserving conditions.     

An algebraic multifrontal preconditioner that exploits the low‐rank property

We present an algebraic structured preconditioner for the iterative solution of large sparse linear systems. The preconditioner is based on a multifrontal variant of sparse LU factorization used with nested dissection ordering. Multifrontal factorization amounts to a partial factorization of a sequence of logically dense frontal matrices, and the preconditioner is obtained if structured factorization is used instead. This latter exploits the presence of low numerical rank in some off‐diagonal blocks of the frontal matrices. An algebraic procedure is presented that allows to identify the hierarchy of the off‐diagonal blocks with low numerical rank based on the sparsity of the system matrix. This procedure is motivated by a model problem analysis, yet numerical experiments show that it is successful beyond the model problem scope. Further aspects relevant for the algebraic structured preconditioner are discussed and illustrated with numerical experiments. The preconditioner is also compared with other solvers, including the corresponding direct solver. Copyright © 2015 John Wiley & Sons, Ltd.     

Tournament matrices and their generalizations,I.

If M is any complex matrix with rank (M + M ^* + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n ? 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + M^T + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n? {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.     

Term rank preservers of Boolean matrices

We obtain some characterizations of linear operators that preserve the term rank of Boolean matrices. That is, a linear operator over Boolean matrices preserves the term rank if and only if it preserves the term ranks 1 and k(≠1) if and only if it preserves the term ranks 2 and l(≠2). Other characterizations of term rank preservers are given.     

Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over ${\mathbb{Q}}$ .     

Cycles of Nonzero Elements in Low Rank Matrices

Dedicated to the memory of Paul Erdős We consider the problem of finding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science. Firstly, the well-known problem on rigidity of matrices, proposed by Valiant as a means to prove lower bounds on some algebraic circuits, is of this type. Secondly, several problems in communication complexity are also of this type. The special case of this problem, where one considers positive semidefinite matrices, is equivalent to the question of arrangements of vectors in euclidean space so that some condition on orthogonality holds. The latter question has been considered by several authors in combinatorics [1, 4]. Furthermore, we can think of this problem as a kind of Ramsey problem, where we study the tradeoff between the rank of the adjacency matrix and, say, the size of a largest complete subgraph. In this paper we show that for an real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles. We get more information for positive semidefinite matrices. Received September 9, 1999 RID="*" ID="*" Partially supported by grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic).     

Products of EP matrices

Necessary and sufficient conditions are given for a matrix to be a product of an EP_r matrix by an EP_s matrix. It is shown that a given square matrix is a product of more than two EP matrices of specified ranks (and hence nullities) if and only if its rank is less than or equal to the minimum of the given ranks and its nullity is less than or equal to the sum of the given nullities. It is also shown that given two EP matrices, the rank of their product is independent of the order of the factors.