William Watkins

Linear maps that preserve commuting pairs of matrices

Abstract Let L be a linear map on the space of n by n matrices with entries in an algebraically closed field of characteristic 0. In this article we characterize all non-singular L with the property that AB = BA implies L ( A ) L ( B ) = L ( B ) L ( A ).

Permanental polynomials of graphs

Abstract The first section surveys recent results on the permanental polynomial of a square matrix A, i.e., per(xI – A). The second section concerns the permanental polynomial of the adjacency matrix of a graph. The final section is an introduction to the permanental polynomial of the Laplacian matrix of a graph. An appendix lists some of these latter polynomials.

Inequalities and identities for generalized matrix functions

Abstract Let χ be a character on the symmetric group S n , and let A = ( a ij ) be an n -by- n matrix. The function d χ ( A ) = Σ σϵS n χ ( σ ) Π n t = 1 a tσ ( t ) is called a generalized matrix function. If χ is an irreducible character, then d χ is called an immanent. For example, if χ is the alternating character, then d χ is the determinant, and if χ ≡ 1, then d χ is called the permanent (denoted per). Suppose that A is positive semidefinite Hermitian. We prove that the inequality (1/χ( id ))d χ (A) ⪕ per A holds for a variety of characters χ including the irreducible ones corresponding to the partitions ( n − 1,1) and ( n − 2,1,1) of n . The main technique used to prove these inequalities is to express the immanents as sums of products of principal subpermanents. These expressions for the immanents come from analogous expressions for Schur polynomials by means of a correspondence of D.E. Littlewood.

Invariants of linear maps on matrix algebras

Let L be a linear map on the space of n×n matrices over a field. We determine the structure of the maps L that preserve commutativity. We also determine the structure of all linear maps on complex matrices that preserve the higher numerical range. The main tool is the Fundamental Theorem of Projective Geometry.

Extremal correlation matrices

Abstract Let R n denote the convex, compact set of all real n -by- n positive semidefinite matrices with main-diagonal entries equal to 1. We examine the extreme points of R n focusing mainly on their rank. the principal result is that R n contains extreme points of rank k if and only if k ( k +1)⩽2 n .

LAPLACIAN UNIMODULAR EQUIVALENCE OF GRAPHS

Let G be a graph. Let L(G) be the difference of the diagonal matrix of vertex degrees and the adjacency matrix. This note addresses the number of ones in the Smith Normal Form of the integer matrix L(G).

The laplacian matrix of a graph: unimodular congruence

This paper gives graph-theoretic conditions that are sufficient for the Laplaceman matrices of two graphs to be congruent by a unimodular matrix.

NOTES ON D-OPTIMAL DESIGNS

Abstract The purpose of this paper is to exhibit new infinite families of D-optimal (0, 1)-matrices. We show that Hadamard designs lead to D-optimal matrices of size ( j , mj ) and ( j − 1, mj ), for certain integers j ≡ 3 (mod 4) and all positive integers m . For j a power of a prime and j ≡ 1 (mod 4), supplementary difference sets lead to D-optimal matrices of size ( j , 2mj ) and ( j − 1, 2mj ), for all positive integers m . We also show that for a given j and d sufficiently large, about half of the entries in each column of a D-optimal matrix are ones. This leads to a new relationship between D-optimality for (0, 1)-matrices and for (±1)-matrices. Known results about D-optimal (±1)-matrices are then used to obtain new D-optimal (0, 1)-matrices.

D-optimal weighing designs for four and five objects

For j = 4 and j = 5 and all d j, the maximum value of detXX , where X runs through all j d (0,1)-matrices, is determined along with a matrix X0 for which the maximum determinant is attained. In the theory of statistical designs, X0 is called a D-optimal design matrix. Design matrices that were previously thought to be D-optimal, are shown here to be D-optimal.

Linear maps and tensor rank

Abstract Let V 1 , …, V m be inner product spaces and A a linear operator on V 1 ⊗ ··· ⊗ V m . Suppose that an equation involving A holds for all tensors of a given rank. Does it follow that the equation holds for all tensors ? We answer this question for some equations involving the inner production V 1 ⊗ ··· ⊗ V m . For example, it is shown that if the field is the complex numbers and ( At , t ) = 0, for all decomposable tensors t , then ( At , t ) = 0, for all tensors t . Thus, A = 0.