On diameter of the commuting graph of a full matrix algebra over a finite field

It is shown that the commuting graph of a matrix algebra over a finite field has diameter at most five if the size of the matrices is not a prime nor a square of a prime. It is further shown that the commuting graph of even-sized matrices over finite field has diameter exactly four. This partially proves a conjecture stated by Akbari, Mohammadian, Radjavi, and Raja [Linear Algebra Appl. 418 (2006) 161–176].     

A new infinite family of minimally nonideal matrices

Minimally nonideal matrices are a key to understanding when the set covering problem can be solved using linear programming. The complete classification of minimally nonideal matrices is an open problem. One of the most important results on these matrices comes from a theorem of Lehman, which gives a property of the core of a minimally nonideal matrix. Cornuéjols and Novick gave a conjecture on the possible cores of minimally nonideal matrices. This paper disproves their conjecture by constructing a new infinite family of square minimally nonideal matrices. In particular, we show that there exists a minimally nonideal matrix with r ones in each row and column for any r?3.     

All group‐based latin squares possess near transversals

In a latin square of order n, a near transversal is a collection of n ?1 cells which intersects each row, column, and symbol class at most once. A longstanding conjecture of Brualdi, Ryser, and Stein asserts that every latin square possesses a near transversal. We show that this conjecture is true for every latin square that is main class equivalent to the Cayley table of a finite group.     

A research problem:

We present a conjecture which when true would generalize T. Ando's characterization of the numerical radius of (bounded linear) operators on a Hilbert space (see [A]). Some evidence for the validity of the conjecture is given. In the finite dimensional case we shall restate the conjecture in terms of convex matrix sets and norms on matrices that are invariant under unitary similarities (u.s.i. norms).     

On a class of matrices generated by certain generalized permutation matrices and applications

ABSTRACT

In this paper, we study a particular class of matrices generated by generalized permutation matrices corresponding to a subgroup of some permutation group. As applications, we first present a technique from which we can get closed formulas for the roots of many families of polynomial equations with degree between 5 and 10, inclusive. Then, we describe a tool that shows how to find solutions to Fermat's last theorem and Beal's conjecture over the square integer matrices of any dimension. Finally, simple generalizations of some of the concepts in number theory to integer square matrices are presented.     

Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields

We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed.     

～A0型箭图的一个表示的自同态代数的Cartan矩阵

为了探讨代数的Cartan矩阵的某些性质与代数分类的关系,通过研究完全域k上的A0型仿射箭图的一个有限维表示的自同态代数的结构与Jordan标准型的关系,并利用Jorelan标准型的组合信息得到了该自同态代数的Cartan矩阵,验证了Cartan矩阵猜想在此情形下不成立.最后提出了一个有关仿射箭图性质的猜想.     

Enumeration of symmetry classes of alternating sign matrices and characters of classical groups

An alternating sign matrix is a square matrix with entries 1, 0 and −1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinant and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round partition functions in terms of irreducible characters of classical groups. In particular, we settle a conjecture on the number of vertically and horizontally symmetric alternating sign matrices (VHSASMs). Dedicated to the memory of David Robbins.     

An algorithm for finding extremal polytope norms of matrix families

In this paper the problem of the computation of the joint spectral radius of a finite set of matrices is considered. We present an algorithm which, under some suitable assumptions, is able to check if a certain product in the multiplicative semigroup is spectrum maximizing. The algorithm proceeds by attempting to construct a suitable extremal norm for the family, namely a complex polytope norm. As examples for testing our technique, we first consider the set of two 2-dimensional matrices recently analyzed by Blondel, Nesterov and Theys to disprove the finiteness conjecture, and then a set of 3-dimensional matrices arising in the zero-stability analysis of the 4-step BDF formula for ordinary differential equations.     

On eigenvalues and equivalent transformation of trigonometric matrices

This paper discusses some issues related to trigonometric matrices arising from the design of finite impulse response (FIR) digital filters. A conjecture on the eigenvalues of a trigonometric matrix is posed with a partial proof given. A new result is also presented on the related equivalent transformation of this trigonometric matrix into a diagonal matrix.     

Products of exponentials of hermitian and complex symmetric matrices

A conjecture involving exponentials of Hermitian matrices is stated, and proved when one of the matrices has rank at most one. As a consequence, the complete conjecture is proved when the matrices are 2×2. A second conjecture involving exponentials of complex symmetric matrices is also stated, and completely proved when the matrices are 2×2. The two conjectures have similar structures but require quite different techniques to analyze.     

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.     

Singular linear systems of differential equations with delays

The differential equation Ax + Bx = Cx(t-l) + f is studied where A, B, C are square matrices. All matrices are allowed to be singular. Examples are given to show that not all initial functions are consistent and that for consistent initial conditions, solutions may be continuous for only a finite time period. Solutions are given explicitly by a recursion formula and consistent initial conditions are explicitly characterized     

New results on the conjecture of Rhodes and on the topological conjecture

The Conjecture of Rhodes, originally called the “type II conjecture” by Rhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems on finite semigroups. It was known that the conjecture of Rhodes is a consequence of another conjecture on the finite group topology for the free monoid. In this paper, we show that the topological conjecture and the conjecture of Rhodes are both equivalent to a third conjecture and we prove this third conjecture in a number of significant particular cases.     

Geršgorin type theorems for quaternionic matrices

This paper aims to set an account of the left eigenvalue problems for real quaternionic (finite) matrices. In particular, we will present the Geršgorin type theorems for the left (and right) eigenvalues of square quaternionic matrices. We shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues of quaternionic matrices.     

Irreducibility Criterion for Quasiregular Representations of the Group of Finite Upper Triangular Matrices

An analog of the quasiregular representation is defined for the group of infinite-order finite upper triangular matrices. It uses G-quasi-invariant measures on some G-spaces. The criterion for the irreducibility and equivalence of the constructed representations is given. This criterion allows us to generalize Ismagilov's conjecture on the irreducibility of an analog of regular representations of infinite-dimensional groups.     

On the complete pivoting conjecture for Hadamard matrices: further progress and a good pivots property

Further progress is achieved for the growth conjecture for Hadamard matrices. It is proved that the leading principal minors of a CP Hadamard matrix form an increasing sequence. Bounds for the sixth and seventh pivot of any CP Hadamard matrix are given. A new proof demonstrating that the growth of a Hadamard matrix of order 12 is 12, is presented. Moreover, a new notion of good pivots is introduced and its importance for the study of the growth problem for CP Hadamard matrices is examined. We establish that CP Hadamard matrices with good pivots satisfy Cryer’s growth conjecture with equality, namely their growth factor is equal to their order. A construction of an infinite class of Hadamard matrices is proposed.     

Alternating sign matrices and descending plane partitions

An alternating sign matrix is a square matrix such that (i) all entries are 1, ?1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math.53 (1979), 193–225) have been discovered, but attempts to prove the existence of such a connection have been unsuccessful. This evidence, however, did suggest a method of proving the Andrews conjecture on descending plane partitions, which in turn suggested a method of proving the Macdonald conjecture on cyclically symmetric plane partitions (Invent. Math.66 (1982), 73–87). In this paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented.     

矩阵的多行拟相合分类及其应用

本文把方阵的拟相合概念推广为长方阵的多行拟相合概念，并对最简情形即Ｍ２，４（Ｆｐ）给出分类结果．此概念之一应用出现在有限环的同构分类之中．