Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

Possible numbers of ones in 0–1 matrices with a given rank

We determine the possible numbers of ones in a 0–1 matrix with given rank in the generic case and in the symmetric case. There are some unexpected phenomena. The rank 2 symmetric case is subtle.     

Possible numbers of ones in 0-1 matrices with a given rank

We determine the possible numbers of ones in a 0-1 matrix with given rank in the generic case and in the symmetric case. There are some unexpected phenomena. The rank 2 symmetric case is subtle.     

Bidiagonalization of oscillatory matrices

We prove that an oscillatory matrix is similar to a bidiagonal nonnegative matrix by means of a totally positive matrix of change of basis. New characterizations of oscillatory and nonsingular totally positive matrices in terms of similarity are provided.     

Rank reduction, factorization and conjugation

We investigate the rank reduction procedure, related factorizations and conjugation algorithms. An exact characterization of the full rank factorization produced by the rank reduction algorithm is given. This result is then used to derive matrix decompositions and conjugation procedures.     

Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.     

三个矩阵秩不等式的多种证明

针对线性代数课程教学中三个矩阵秩的不等式的证明展开研究,对每一个不等式,分别给出相应的证明。     

The minimal polynomial of a matrix

Let Rbe a finite dimensional central simple algebra over a field FA be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A(λ) of A over F. By using q_A(λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

The minimal polynomial of a matrix ∗

Let Rbe a finite dimensional central simple algebra over a field F A be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A (λ) of A over F. By using q_A (λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

Perturbation theory for the Eckart-Young-Mirsky theorem and the constrained total least squares problem

Golub et al. (Linear Algebra Appl. 88/89 (1987) 317–327), J.Demmel (SIAM J. Numer. Anal. 24 (1987) 199–206), generalized the Eckart-Young-Mirsky (EYM) theorem, which solves the problem of approximating a matrix by one of lower rank with only a specific rectangular subset of the matrix allowed to be changed. Based on their results, this paper presents perturbation analysis for the EYM theorem and the constrained total least squares problem (CTLS).     

Products of symmetric and skew-symmetric Matrices

The class of matrices which can be represented as products of two matrices, each of which is either symmetric or skew-symmetric, is identified. Possible ranks of the factors in such representations of a given matrix are identified as well.     

Products of symmetric and skew-symmetric Matrices ∗

The class of matrices which can be represented as products of two matrices, each of which is either symmetric or skew-symmetric, is identified. Possible ranks of the factors in such representations of a given matrix are identified as well.     

矩阵环F[A]中元素的可逆性

研究了矩阵环F[A]中元素可逆的条件,讨论了矩阵环F[A]上的矩阵的初等变换与初等矩阵的性质,给出了求F[A]中可逆元的逆元的一个简便方法.     

The Minimum Rank of a 3 × 3 Partial Block Matrix

After recalling the definition and some basic properties of finite hypergroups—a notion introduced in a recent paper by one of the authors—several non-trivial examples of such hypergroups are constructed. The examples typically consist of n n×n matrices, each of which is an appropriate polynomial in a certain tri-diagonal matrix. The crucial result required in the construction is the following: ‘let A be the matrix with ones on the super-and sub-diagonals, and with main diagonal given by a ₁…a _n which are non-negative integers that form either a non-decreasing or a symmetric unimodal sequence; then A_k =P_k (A) is a non-negative matrix, where p_k denotes the characteristic polynomial of the top k× k principal submatrix of A, for k=1,…,n. The matrices A_k as well as the eigenvalues of A, are explicitly described in some special cases, such as (i) a_i =0 for all ior (ii) a_i =0 for i<n and a_n =1. Characters ot finite abelian hypergroups are defined, and that naturally leads to harmonic analysis on such hypergroups.     

Some rank equalities and inequalities for Kronecker products of matrices

A set of rank equalities and inequalities are established for block matrices consisting of Kronecker products. Various consequences are also given.     

Some rank equalities and inequalities for Kronecker products of matrices

A set of rank equalities and inequalities are established for block matrices consisting of Kronecker products. Various consequences are also given.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

Sensitivity analysis of nondifferentiable sums of singular values of rectangular matrices

Let K(x) be an s-by-r rectangular matrix depending on a parameter x ε E and denote by g(x) the sum of its m largest singular values (1 ≤ m ≤ Min{s,r}). If K(x) depends affinely on x, then g is a nondifferentiable convex function. In this paper we consider first the affine case and give some formulas for the conjugate, subdifferential, and ε-subdifferential of g. These formulas are then used to obtain perturbation bounds for g(x). We study next the nonaffine case and discuss some questions related with the regularity, generalized subdifferentiability, and directional differentiability of g.     

Ranks of Solutions of the Matrix Equation AXB = C

The purpose of this article is to solve two problems related to solutions of a consistent complex matrix equation AXB = C : (I) the maximal and minimal ranks of solution to AXB = C , and (II) the maximal and minimal ranks of two real matrices X₀ and X₁ in solution X = X₀ + iX₁ to AXB = C . As applications, the maximal and minimal ranks of two real matrices C and D in generalized inverse (A + iB)^- = C + iD of a complex matrix A + iB are also examined.