Infinite triangular matrices, q-Pascal matrices, and determinantal representations

We use basic properties of infinite lower triangular matrices and the connections of Toeplitz matrices with generating-functions to obtain inversion formulas for several types of q-Pascal matrices, determinantal representations for polynomial sequences, and identities involving the q-Gaussian coefficients. We also obtain a fast inversion algorithm for general infinite lower triangular matrices.     

Canonical forms for normal matrices that commute with their complex conjugate

We consider the class of normal complex matrices that commute with their complex conjugate. We show that such matrices are real orthogonally similar to a canonical direct sum of 1-by-1 and certain 2-by-2 matrices. A canonical form for quasi-real normal matrices is obtained as a special case. We also exhibit a special form of the spectral theorem for normal matrices that commute with their conjugate.     

Intrinsic products and factorizations of matrices

We say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates. Analogously we speak about intrinsic product of two or more matrices, as well as about intrinsic factorizations of matrices. Since all entries of the intrinsic product are products of entries of the multiplied matrices, there is no addition. We present several examples, together with important applications. These applications include companion matrices and sign-nonsingular matrices.     

Dominant matrices and max algebra

We study the class of so-called totally dominant matrices in the usual algebra and in the max algebra in which the sum is the maximum and the multiplication is usual. It turns out that this class coincides with the well known class of positive matrices having positive the determinants of all 2×2 submatrices. The closure of this class is closed not only with respect to the usual but also with respect to the max multiplication. Further properties analogous to those of totally positive matrices are proved and some connections to Monge matrices are mentioned.     

Singular, nonsingular, and bounded rank completions of ACI-matrices

An affine column independent matrix is a matrix whose entries are polynomials of degree at most 1 in a number of indeterminates where no indeterminate appears with a nonzero coefficient in two different columns. A completion is a matrix obtained by giving values to each of the indeterminates. Affine column independent matrices are more general than partial matrices where each entry is either a constant or a distinct indeterminate. We determine when the rank of all completions of an affine column independent matrix is bounded by a given number, generalizing known results for partial matrices. We also characterize the square partial matrices over a field all of whose completions are nonsingular. The maximum number of free entries in such matrices of a given order is determined as well as the partial matrices with this maximum number of free entries.     

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.     

Interlacing properties of the eigenvalues of some matrix classes

We establish the eigenvalue interlacing property (i.e. the smallest real eigenvalue of a matrix is less than the smallest real eigenvalue of any of its principal submatrices) for the class of matrices introduced by Kotelyansky (all principal and almost principal minors of these matrices are positive). We show that certain generalizations of Kotelyansky and totally positive matrices possess this property. We also prove some interlacing inequalities for the other eigenvalues of Kotelyansky matrices.     

An extremal sparsity property of the Jordan canonical form

We prove that among all the matrices that are similar to a given square complex matrix, the Jordan canonical form has the largest number of off-diagonal zero entries. We also characterize those matrices that attain this largest number.     

The α-scalar diagonal stability of block matrices

We generalize two results: Kraaijevanger’s 1991 characterization of diagonal stability via Hadamard products and the block matrix version of the closure of the positive definite matrices under Hadamard multiplication. We restate our generalizations in terms of P^α-matrices and α-scalar diagonally stable matrices.     

ACI-matrices all of whose completions have the same rank

We characterize the ACI-matrices all of whose completions have the same rank, determine the largest number of indeterminates in such partial matrices of a given size, and determine the partial matrices that attain this largest number.     

How real is your matrix?

For scalars there is essentially just one way to define reality, real part and to measure nonreality. In this paper various ways of defining respective concepts for complex-entried matrices are considered. In connection with this, products of circulant and diagonal matrices often appear and algorithms to approximate additively and multiplicatively with them are devised. Multiplicative structures have applications, for instance, in diffractive optics, preconditioning and fast Fourier expansions.     

A matrix trace inequality and its application

In this short paper, we give a complete and affirmative answer to a conjecture on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson inequality for positive semidefinite matrices.     

Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms

A set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product. The extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research. In this article, we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set, but also for all sufficiently nearby sets of equal cardinality. We also prove a theorem giving conditions under which the Barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar, and show that in certain cases these conditions are also persistent under small perturbations.     

Efficiently computing the permanent and Hafnian of some banded Toeplitz matrices

We present a new efficient method for computing the permanent and Hafnian of certain banded Toeplitz matrices. The method covers non-trivial cases for which previous known methods do not apply. The main idea is to use the elements of the first row and column, which determine the entire Toeplitz matrix, to construct a digraph in which certain paths correspond to permutations that the permanent and Hafnian count. Since counting paths can be done efficiently, the permanent and Hafnian for those matrices is easily obtainable.     

The base sets of primitive zero-symmetric sign pattern matrices

In [J.Y. Shao, L.H. You, Bound on the base of irreducible generalized sign pattern matrices, Discrete Math., in press], Shao and You extended the concept of the base from powerful sign pattern matrices to non-powerful (and generalized) sign pattern matrices. In this paper, we study the bases of primitive zero-symmetric sign pattern (and generalized sign pattern) matrices. Sharp upper bounds of the bases are obtained. We also show that there exist no “gaps” in the base sets of the classes of such matrices.     

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     

Determinant of the distance matrix of a tree with matrix weights

Let T be a tree with n vertices and let D be the distance matrix of T. According to a classical result due to Graham and Pollack, the determinant of D is a function of n, but does not depend on T. We allow the edges of T to carry weights, which are square matrices of a fixed order. The distance matrix D of T is then defined in a natural way. We obtain a formula for the determinant of D, which involves only the determinants of the sum and the product of the weight matrices.     

Cayley transform and the Kronecker product of Hermitian matrices

We consider the conditions under which the Cayley transform of the Kronecker product of two Hermitian matrices can be again presented as a Kronecker product of two matrices and, if so, if it is a product of the Cayley transforms of the two Hermitian matrices. We also study the related question: given two matrices, which matrix under the Cayley transform yields the Kronecker product of their Cayley transforms.     

Bijective linear rank preservers for spaces of matrices over antinegative semirings

We classify the bijective linear operators on spaces of matrices over antinegative commutative semirings with no zero divisors which preserve certain rank functions such as the symmetric rank, the factor rank and the tropical rank. We also classify the bijective linear operators on spaces of matrices over the max-plus semiring which preserve the Gondran-Minoux row rank or the Gondran-Minoux column rank.     

Some counting questions for matrices with restricted entries

We obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box (in an arbitrary location) and a given determinant. We then apply these bounds to estimate the number of matrices in such boxes which have an integer eigenvalues. Finally, we outline some open questions.