Integer Powers of Anti-Bidiagonal Hankel Matrices

In this paper we derive a general expression for integer powers of real upper and lower anti-bidiagonal matrices with constant anti-diagonals using Chebyshev polynomials. An explicit formula for the inverse of these matrices is also provided.     

Sign Regular Matrices of Order Two

A matrix is called sign regular of order k if every minor of order i has the same sign for each i = 1,2,<, k . If an m × n matrix is sign regular of order k for k = min { m,n } then it is called sign regular. This paper studies some properties of sign regular matrices of order two. Remarkable properties are proved when the row sums of these matrices form a monotone vector.     

A Generalization of Alternating Sign Matrices

In alternating sign matrices, the first and last nonzero entry in each row and column is specified to be +1. Such matrices always exist. We investigate a generalization by specifying independently the sign of the first and last nonzero entry in each row and column to be either a +1 or a ?1. We determine necessary and sufficient conditions for such matrices to exist whose proof contains an algorithm for their construction.     

Convolution Powers of Probabilities on Stochastic Matrices

For any probability on the space of d×d stochastic matrices we associate a probability ; on a finite group—a subgroup of the permutation group—related to the kernel of the semigroup generated by the support of . We show that ⁿ converges iff ⁿ converges.     

Pattern Avoidance in Alternating Sign Matrices

We generalize the definition of a pattern from permutations to alternating sign matrices. The number of alternating sign matrices avoiding 132 is proved to be counted by the large Schr?der numbers, 1, 2, 6, 22, 90, 394, .... We give a bijection between 132-avoiding alternating sign matrices and Schr?der paths, which gives a refined enumeration. We also show that the 132-, 123-avoiding alternating sign matrices are counted by every second Fibonacci number. Received January 2, 2007     

The Module Decomposition of Super-Symmetric Powers of Matrices

ABSTRACT

Let M be the k  ×  m matrices over ?. The GL ( k ) ×  GL ( m ) decompositions of the symmetric and of the exterior powers of M are described by two classical theorems. We describe a theorem for Lie superalgebras, which implies both of these classical theorems as special cases. The constructions of both the exterior and the symmetric algebras are generalized to a class of algebras defined by partitions. That superalgebra theorem is further generalized to these algebras.     

坡上矩阵的逆与幂(英文)

Invertible conditions,indices and periods of matrices over an incline are considered.In all proofs multiplicative commutativity is not used.Some results about invertibility, indices and periods in a commutative incline are proved to be correct also for any incline.     

Additive Decomposition of Real Matrices

The orthogonal orbit ${\cal O}(A)$ of an n × n real matrix A is the set of real matrices of the form $P^t \ AP$ where $P^t P = I_n$ . We show that $A/ \| A\|$ is an affine sum of four orthogonal matrices, and note that $A^t$ can always be written as an affine combination of no more than 2 n m 1 matrices in ${\cal O}(A)$ . This improves some recent results of Zhan, and answers some of his questions. Other related results are also discussed.     

Weighted Counting of Inversions on Alternating Sign Matrices

We generalize the author’s formula (2011) on weighted counting of inversions on permutations to one on alternating sign matrices. The proof is based on the sequential construction of alternating sign matrices from the unit matrix which essentially follows from the earlier work of Lascoux-Schützenberger (1996).

     

k次幂非正的符号模式矩阵

本首先对使得A^K≤0的符号模式矩阵A进行了刻画（k为任意正整数），进而决定了这类矩阵中负元个数的最大值。最后给出了使得A^2≤0的符号模式矩阵A的充分必要条件。     

Genus Expansion for Real Wishart Matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central-limit-type theorem for the asymptotic distribution around the expected value.     

关于实方阵的正定性

本文研究一般实方阵的正定性 ,给出了方阵正定的一些充分必要条件     

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant matrices. A. Massey’s current address: Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. e-mail: amassey3102@math.ucla.edu.     

实正定矩阵的复合矩阵的正定性

本文讨论了实正定矩阵的复合矩阵的正定性，并且给出了实正定矩阵的复合矩阵仍为正定矩阵的一个充要条件.     

Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas

An alternating sign matrix is a square matrix whose entries are 1, 0, or –1, and which satisfies certain conditions. Permutation matrices are alternating sign matrices. In this paper, we use the (generalized) Littlewood's formulas to expand the products and 2 as sums indexed by sets of alternating sign matrices invariant under a 180° rotation. If we put t = 1, these expansion formulas reduce to the Weyl's denominator formulas for the root systems of type B _n and C _n. A similar deformation of the denominator formula for type D _n is also given.     

On Sign Conditions Over Real Multivariate Polynomials

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions over the polynomials. In addition, we extend these results to the case of closed sign conditions over an arbitrary family of real multivariate polynomials. The complexity bounds for these procedures improve the known ones.     

关于实方阵的几个问题

本文讨论如下内容：１．把有关对称正定（半正定）的一些性质推广到广义正定（半正定）。２．给定ｘ∈Ｒｍ×ｍ，∧为对角阵，求ＡＸ＝ｘ∧在对称半正定矩阵类中解存在的充要条件及一般形式，并讨论了对任意给定的对称正定（半正定）矩阵Ａ，在上述解的集合中求得Ａ，使得     

基于实Schur分解的实正规矩阵特征值的变分特征

在实Schur分解的基础上,构造一新特征量表示正规矩阵特征值的虚部最大值,同时表示了所有实部.