On a characterization of irreducibility of a non-negative matrix

Let A denote an n×n matrix with all its elements real and non-negative, and let r_i be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r₁,…,r_n), where D(r₁,…,r_n) is the diagonal matrix with r_i at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of B^T contains a vector d whose entries are all non-null.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

Continuity properties of the Drazin pseudoinverse

Let B^D denote that Drazin inverse of the n×n complex matrix B. Define the core-rank of B as rank (B^i(B)) where i(B) is the index of B. Let j = 1,2,…, and A_j and A be square matrices such that A_i converges to A with respect to some norm. The main result of this paper is that A_j^D converges to A^D if and only if there exist a j₀ such that core-rank A_j=core-rankA for j ? j₀.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

The inverse M-matrix problem

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(a_ij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? a_ij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A^?1=B=(b_ij), then b_ii> 0 and b_ij ? 0 for i≠_j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x²=sy+(1?s)y²; then B is an M-matrix if s^?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for a_ij=a_jj=x when i=n?1, n and 1 ? _j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix.     

Polyboxes,Cube Tilings and Rigidity

A non-empty subset A of X=X ₁×???×X _d is a (proper) box if A=A ₁×???×A _d and A _i ?X _i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A _i =B _i , A _i =X _i ?B _i , A _i ?{B _i ,X _i ?B _i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A _i =X _i ?B _i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.     

Sieve-equivalence and explicit bijections

Suppose A₁,…, A_n are subsets of a finite set A, and B₁,…, B_n are subsets of a finite set B. For each subset S of N = {1, 2,…, n}, let A_s = ∩_i?SA_i and B_S = ∩_i?SB_i. It is shown that if explicit bijections f_S:A_S → B_S for each S ? N are given, an explicit bijection h:A-∪_i=1A_i→B-∪_i=1B_i can be constructed. The map h is independent of any ordering of the elements of A and B, and of the order in which the subsets A_i and B_i are listed.     

r-Numbers completion problems of partial upper canonical form I

A pair (A, B), where A is an n × n matrix and B is an n × m matrix, is said to have the nonnegative integers sequence {r_j}_j=1^p as the r-numbers sequence if r₁ = rank(B) and r_j = rank[B AB … A^j−1 B] − rank[B AB … A^j−2B], 2 ≤ j ≤ p. Given a partial upper triangular matrix A of size n × n in upper canonical form and an n × m matrix B, we develop an algorithm that obtains a completion A_c of A, such that the pair (A^c, B) has an r-numbers sequence prescribed under some restrictions.     

Small divisors of Bernoulli sums

Let ?= {?_i,i ≥1} be a sequence of independent Bernoulli random variables (P{?_i = 0} = P{?_i = 1 } = 1/2) with basic probability space (Ω, A, P). Consider the sequence of partial sums B_n=?₁+...+?_n, n=1,2..... We obtain an asymptotic estimate for the probability P{P^-(B_n) > >} for >≤n^{e/log log n}, c a positive constant.     

L2(Rs)中正交尺度函数和正交小波的Fourier变换紧支集的刻画

借助于Fourier变换,在较弱条件下给出了φ(x)是L²(R^s)上正交尺度函数的一个充分必要条件.进一步, 假设 {Ψ_μ } 是正交小波, 且正交小波的Fourier变换紧支集是 ∪_μsupp{ψ_μ} =∏^s_i=1[A_i, D_i] -∏^s_i=1(B_i, C_i),A_i≤B_i≤C_i≤D_i, i =1, 2,… , s. 则在最弱条件“每一个 |ψ_μ| 在ω∈∂(∏^s_i=1[A_i, D_i]) 上连续'下, 该文通过一些不等式和等式给出了正交尺度函数和正交小波的Fourier变换紧支集的刻画.文中的结论全面改进了龙瑞麟和张之华的结果.     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

A generalization of Lavoie's inequality concerning the sum of idempotent matrices

Let a complex pxn matrix A be partitioned as A′=(A′₁,A′₂,…,A′_k). Denote by ?(A), A′, and A^? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A⁽¹⁴⁾ be an inner inverse of A such that A⁽¹⁴⁾A is a Hermitian matrix. Let B=(A⁽¹⁴⁾₁,A⁽¹⁴⁾₂,…,A_k⁽¹⁴⁾) and $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$.Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A⁽¹⁴⁾ or $\text{A}{}_{\mathrm{i>}}\text{A}{}_{\mathrm{<mtext>j</mtext>}}{}^1\text{=0}$ for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$ is no longer true.     

An extended Kronecker product of matrices

Let $\text{A = (A}{}_1\text{¦ A}{}_2\text{¦ ··· ¦ A}{}_{\mathrm{r}}\text{)}\text{and}\text{B = (B}{}_1\text{¦ B}{}_2\text{¦ ··· ¦ B}{}_{\mathrm{r}}\text{)}$ be column-wise partitioned matrices over complex numbers. Then an extended Kronecker product is $\text{A ⊙ B = (A}{}_1\text{? B}{}_1\text{¦ ··· ¦ A}{}_{\mathrm{r}}\text{? B}{}_{\mathrm{r}}\text{)}$, where A_i ? B_i is the Kronecker product of A_i and B_i. Some properties of an extended Kronecker product of matrices are investigated. The properties of the solutions of the systems of linear equations whose coefficient matrices are extended Kronecker products of matrices are studied.     

More bounds for elgenvalues using traces

Let the n × n complex matrix A have complex eigenvalues λ₁,λ₂,…λ_n. Upper and lower bounds for Σ(Reλ_i)² are obtained, extending similar bounds for Σ|λ_i|² obtained by Eberlein (1965), Henrici (1962), and Kress, de Vries, and Wegmann (1974). These bounds involve the traces of A¹A, B², C², and D², where $\text{B}\text{=}\text{1}\text{2}\text{(}\text{A}\text{+}\text{A}{}^1\text{)}$, $\text{C}\text{=}\text{1}\text{2}\text{(}\text{A}\text{?}\text{A}{}^1\text{) /i}$, and $\text{D}\text{=}\text{AA}{}^1\text{?}\text{A}{}^1\text{A}$, and strengthen some of the results in our earlier paper “Bounds for eigenvalues using traces” in Linear Algebra and Appl. [12].     

Permanental pairs of doubly stochastic matrices. II

The n ×n doubly stochastic matrices A, B form a permanental pair if the permanent of every convex linear combination λA+(1?λ)B(0?λ?1) is independent of λ A, B are called mates. In this article we show that the direct sum of any number, k, of matrices J_i (of varying individual dimension) cannot have a mate. Here J_i is the n_i×n_i matrix with every entry equal to $\text{1}\text{n}{}_{\mathrm{i}}$;∑n_i=n.     

On Strongly Clean Modules

An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A ₁⊕ A ₂ with M′? M, there are decompositions M′ = M ₁⊕ M ₂, B = B ₁⊕ B ₂, and A _i  = C _i ⊕ D _i (i = 1,2) such that M ₁⊕ B ₁ = C ₁⊕ D ₂ = M ₁⊕ C ₁ and M ₂⊕ B ₂ = D ₁⊕ C ₂ = M ₂⊕ C ₂.     

两类惯量惟一的对称符号模式

§ 1　 IntroductionA sign pattern(matrix) A is a matrix whose entries are from the set{ +,-,0 } .De-note the setofall n× n sign patterns by Qn.Associated with each A=(aij)∈ Qnis a class ofreal matrices,called the qualitative class of A,defined byQ(A) ={ B =(bij)∈ Mn(R) |sign(bij) =aijfor all i and j} .　　 For a symmetric sign pattern A∈ Qn,by G(A) we mean the undirected graph of A,with vertex set { 1 ,...,n} and (i,j) is an edge if and only if aij≠ 0 .A sign pattern A∈ Qnis a do…     

Matrix extensions of Liouville-Dirichlet-type integrals

The Dirichlet integral provides a formula for the volume over the k-dimensional simplex ω={x₁,…,x_k: x_i?0, i=1,…,k, s?∑^k₁x_i?T}. This integral was extended by Liouville. The present paper provides a matrix analog where now the region becomes $\text{Ω={V}{}_1\text{,…,V}{}_{\mathrm{k}}\text{: V}{}_{\mathrm{i}}\text{>0, i=1,…,k, 0?∑V}{}_{\mathrm{i}}\text{?t}}$, where now each V_i is a p×p symmetric matrix and A?B means that A?B is positive semidefinite.