Continuity and separation in symmetric topologies

The well-known Frobenius rank inequality established by Frobenius in 1911 states that the rank of the product ABC of three matrices satisfies the inequality rank(ABC) U rank(AB) + rank(BC)- rank(B) A new necessary and sufficient condition for equality to hold is presented and then some interesting consequences and applications are discussed.     

Products of elementary doubly stochastic matrices

While studying a theorem of Westwerk on higher numerical ranges, we became interested in how the theory of elementary doubly stochastic (e.d.s.) matrices is related to a result of Goldberg and Straus. We show that there exist classes of doubly stochastic (d.s.) matrices of order n≧3 and orthostochastic (o s) matrices of order n≧4 such that the matrices in these classes cannot be represented as a product of e.d.s. matrices. In fact the matrices in these classes do not admit a representation as an infinite limit of a product of e.d.s. matrices.     

三个矩阵乘积厄米特Moore-Penrose逆的等式与不等式

We investigate relationships between the Moore-Penrose inverse(ABA*)and the product [(AB)(1,2,3)]*B(AB)(1,2,3)through some rank and inertia formulas for the difference of(ABA*)-[(AB)(1,2,3)]*B(AB)(1,2,3),where B is Hermitian matrix and(AB)(1,2,3)is a {1,2,3}-inverse of AB.We show that there always exists an(AB)(1,2,3)such that(ABA*)= [(AB)(1,2,3)]*B(AB)(1,2,3)holds.In addition,we also establish necessary and sufficient conditions for the two inequalities(ABA*) [(AB)(1,2,3)]*B(AB)(1,2,3)and(ABA*)[(AB)(1,2,3)]*B(AB)(1,2,3)to hold in the L¨owner partial ordering.Some variations of the equalities and inequalities are also presented.In particular,some equalities and inequalities for the Moore-Penrose inverse of the sum A + B of two Hermitian matrices A and B are established.     

Every Matrix is a Product of Toeplitz Matrices

We show that every \(n\,\times \,n\) matrix is generically a product of \(\lfloor n/2 \rfloor + 1\) Toeplitz matrices and always a product of at most \(2n+5\) Toeplitz matrices. The same result holds true if the word ‘Toeplitz’ is replaced by ‘Hankel,’ and the generic bound \(\lfloor n/2 \rfloor + 1\) is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not, in general, replace the subspace of Toeplitz or Hankel matrices by an arbitrary \((2n-1)\)-dimensional subspace of \({n\,\times \,n}\) matrices. Furthermore, such decompositions do not exist if we require the factors to be symmetric Toeplitz or persymmetric Hankel, even if we allow an infinite number of factors.     

Explicit invariant measures for products of random matrices

We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL. The matrices in the product are such that one entry is gamma-distributed along a ray in the complex plane. When the ray is the positive real axis, the products are those associated with a continued fraction studied by Letac & Seshadri [Z. Wahr. Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the continued fraction is a generalised inverse Gaussian. We extend this result by finding the distribution for an arbitrary ray in the complex right-half plane, and thus compute the corresponding Lyapunov exponent explicitly. When the ray lies on the imaginary axis, the matrices in the infinite product coincide with the transfer matrices associated with a one-dimensional discrete Schrödinger operator with a random, gamma-distributed potential. Hence, the explicit knowledge of the Lyapunov exponent may be used to estimate the (exponential) rate of localisation of the eigenstates.

     

Relations between certain matrix products similar to commutativity

In this note it is shown that for certain pairs of (infinite) matrices A,B whose product is not commutative, there holds a relation AB=DA, similar to commutativity in that D is obtained from B by a very slight modification: by the deletion of some rows, or by the deletion of some rows and columns.     

关于矩阵群逆的逆序律

得到了体上两个n阶方阵A,B的群逆A#,B#若存在,则其乘积的群逆(AB) #也存在,且(AB) #=B#A#成立的充分与必要条件是:存在n阶可逆矩阵P使得A =Pdiag(A1,A2 ,…,As) P- 1,B =Pdiag(B1,B2 ,…,Bs) P- 1且对于任意i(i=1 ,2 ,…,s)有Ai,Bi阶数相同,Ai,Bi为可逆矩阵或为0矩阵;又对i≠1有Ai Bi=0 .     

Decomposition of Matrices into Involutory Matrices and Symmetric Matrices

Let \Omega be a field, and let F denote the Frobenius matrix: $[F = \left( {\begin{array}{*{20}{c}} 0&{ - {\alpha _n}}\{{E_{n - 1}}}&\alpha \end{array}} \right)\]$ where \alpha is an n-1 dimentional vector over Q, and E_n- 1 is identity matrix over \Omega. Theorem 1. There hold two elementary decompositions of Frobenius matrix: (i) F=SJB, where S, J are two symmetric matrices, and B is an involutory matrix; (ii) F=CQD, where O is an involutory matrix, Q is an orthogonal matrix over \Omega, and D is a diagonal matrix. We use the decomposition (i) to deduce the following two theorems: Theorem 2. Every square matrix over \Omega is a product of twe symmetric matrices and one involutory matrix. Theorem 3. Every square matrix over \Omega is a product of not more than four symmetric matrices. By using the decomposition (ii), we easily verify the following Theorem 4(Wonenburger-Djokovic') . The necessary and sufficient condition that a square matrix A may be decomposed as a product of two involutory matrices is that A is nonsingular and similar to its inverse A^-1 over Q (See [2, 3]). We also use the decomosition (ii) to obtain Theorem 5. Every unimodular matrix is similar to the matrix CQB, where C, B are two involutory matrices, and Q is an orthogonal matrix over Q. As a consequence of Theorem 5. we deduce immediately the following Theorem 6 (Gustafson-Halmos-Radjavi). Every unimodular matrix may be decomposed as a product of not more than four involutory matrices (See [1] ). Finally, we use the decomposition (ii) to derive the following Thoerem 7. If the unimodular matrix A possesses one invariant factor which is not constant polynomial, or the determinant of the unimodular matrix A is I and A possesses two invariant factors with the same degree (>0), then A may be decomposed as a product of three involutory matrices. All of the proofs of the above theorems are constructive.     

Range invariance of certain matrix products

A necessarv and sufficient condition is established for the product AB C to have its range.A (AB C), invariant with respect to the choice of a generalized inverse B .This result is then used to derive criteria for the invariance of the subspaces A(AB ).A(B C)A(B) and A(BB C) and also to deduce that the simultaneous invariance of the range of AB^- C and the range of its conjugate transpose entails the invariance of the product AB^-C itself.     

The strengthened versions of the additive and multiplicative Weyl inequalities

The strengthened versions of the classical additive and multiplicative Weyl inequalities for the singular values of A + B and AB*, where A and B are rectangular matrices, and for the eigenvalues of A + B and AB, where A and B are Hermitian matrices, are established under certain assumptions on the subspaces spanned by some singular vectors or eigenvectors, respectively, of A and B. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 39–59.     

关于复矩阵乘积的正定性

两复正定矩阵之和必是复正定矩阵,但其积未必是复正定矩阵.研究了复矩阵之积的正定性,给出了复矩阵之积为复正定矩阵的一系列判定条件,获得了一些新的结果,改进并推广了K y Fan T aussky定理及Fe jer定理.     

域上矩阵积的广义逆及自反广义逆的逆反律

令A,B是任意域上的矩阵且使得AB有意义。本文研究了AB的广义逆、自反广义逆与A,B的广义逆、自反广义逆的积之间的关系,得到了B{1}A{1}(AB){1},B{1}A{1}=(AB){1},B{1,2}A{1,2}(AB){1,2}和B{1,2}A{1,2}=(AB){1,2}成立的一些充要条件。     

围长为2的本原无限布尔方阵类的本原指数集

研究了围长为2的无限布尔方阵的本原性,通过无限有向图D（A）的直径给出了这类矩阵的本原指数的上确界,最后证明了直径小于等于d且围长为2的本原无限布尔方阵所构成的矩阵类的本原指数集为Ed^0={2,3,…,3d}．     

差分代换矩阵与多项式的非负性判定

主要分析了差分代换矩阵的基本性质,证明了存在有限个差分代换矩阵的乘积可以将单位点$(1,0,\cdots,0)$变换到指定的非负(本原)整点.利用这一结果可以导出${R^n_+}$上判定半正定型的充要条件.根据此充要条件建立的算法(TSDS)可能不停机,针对不停机的情况,再给出一些判定半正定型的充分条件.     

Optimal norms and the computation of joint spectral radius of matrices

The notion of spectral radius of a set of matrices is a natural extension of spectral radius of a single matrix. The finiteness conjecture (FC) claims that among the infinite products made from the elements of a given finite set of matrices, there is a certain periodic product, made from the repetition of the optimal product, whose rate of growth is maximal. FC has been disproved. In this paper it is conjectured that FC is almost always true, and an algorithm is presented to verify the optimality of a given product. The algorithm uses optimal norms, as a special subset of extremal norms. Several conjectures related to optimal norms and non-decomposable sets of matrices are presented. The algorithm has successfully calculated the spectral radius of several parametric families of pairs of matrices associated with compactly supported multi-resolution analyses and wavelets. The results of related numerical experiments are presented.     

强正定二次型与Weyl-Heisenberg 框架的构造

本文研究了一类具有特殊结构的无限维二次型, 得到这类二次型的对称矩阵是符号为多项式的模的平方的Laurent 矩阵, 进一步得到了这类二次型是强正定的判断标准以及一类Weyl-Heisenberg 框架的构造. 本文还研究了这类二次型的矩阵的所有有限维主对角子矩阵的强正定性, 并由此得到一类子空间Weyl-Heisenberg 框架的构造. 最后举例说明本文的主要结果及其应用. 本文建立了两个看似不相关的领域间的联系.     

Infinite analogues of block toeplitz matrices and related orthogonal functions

This paper concerns a class of infinite block matrices that are analogous to finite block Toeplitz matrices. Also studied are corresponding matrix-valued functions that are orthogonal for a matrixvalued inner product. An appendix presents basic results on orthogonalization in a Hilbert module.     

一类无穷下三角矩阵的代数性质

修正了 [4,5]中的 Jabotinsky矩阵,得到并证明了一类无穷下三角矩阵T（f）的一些性质,最后,导出了一些与导数相关的反演关系和组合恒等式.     

A solution to Schrödinger's problem of non-linear integral equations

A solution of Schrödinger's system of non-linear integral equations determines the rate function of a large deviation principle for kernels with prescribed marginal distributions. This kind of large deviation principle has some meaning in quantum mechanics.Diffusion equations associated with Schrödinger equations have typically transition functions with singular creation and killing. Hence they provide measurable non-negative generally unbounded kernels which may vanish on sets with positive measure and which can possess infinite mass.For Schrödinger systems with such kernels, a solution is proved to exist uniquely in terms of a product measure. It is obtained from a variational principle for the local adjoint of a product measure endomorphism. The generally unbounded factors of the solution are characterized by integrability properties.     

Perron-frobenius-type theorems for tridiagonal operators

It is proved that in a large class of bounded tridiagonal operators (infinite Jacobi matrices), not necessarily positive or non-negative, positive eigenvalues exist and the eigenvector which corresponds to the greatest of them can be taken strictly positive. It is the unique positive eigenvector up to a constant multiple.