Shaun M. Fallat

Sub-direct sums and positivity classes of matrices

Abstract It is well known that a direct sum is positive semidefinite if and only if each of the direct summands is positive semidefinite. In fact, it is also known that this statement remains true if positive semidefinite is replaced with: doubly nonnegative, completely positive, totally nonnegative, M -matrix and P -matrix, etc. For each of these classes we consider corresponding questions for a more general “sum” of two matrices, of which the direct sum and ordinary sum are special cases.

Perron components and algebraic connectivity for weighted graphs

The algebraic connectivity of a connected graph is the second-smallest eigenvalue of its Laplacian matrix, and a remarkable result of Fiedler gives information on the structure of the eigenvectors associated with that eigenvalue. In this paper, we introduce the notion of a perron component at a vertex in a weighted graph, and show how the structure of the eigenvectors associated with the algebraic connectivity can be understood in terms of perron components. This leads to some strengthening of Fiedlers original result, gives some insights into weighted graphs under perturbation, and allows for a discussion of weighted graphs exhibiting tree-like structure.

Maximum determinant of (0,1) matrices with certain constant row and column sums

The maximum absolute value of the determinant of n x n nonsinguiar (0,1) matrices that have constant line sums (i.e., row sums and column sums) k is investigated. For n≠4k=2, this maximum determinant is determined to be 2t if n=3t or 3t+2, and 2t−1 if n=3t+1. Restriction to a subset of these matrices, namely those that are symmetric and have zero trace (their graphs are regular of degree k), leaves the maximum unchanged for k=2. For this restricted class, when n≥7k=n−3, the maximum absolute value of the determinant is (n−3)3[n/4]−-1, This maximum gives a lower bound for the maximum absolute value of the determinant for the larger class, but in general this bound is not tight. Other determinantal values and bounds for specific n and k are given.

EIGENVALUE LOCATION FOR NONNEGATIVE AND Z-MATRICES

Let Lk0 denote the class of n × n Z-matrices A = tl − B with B ⩾ 0 and ϱk(B) ⩽ t < ϱk + 1(B), where ϱk(B) denotes the maximum spectral radius of k × k principal submatrices of B. Bounds are determined on the number of eigenvalues with positive real parts for A ϵ Lk0, where k satisfies, ⌊n2⌋ ⩽ k ⩽ n − 1. For these classes, when k = n − 1 and n − 2, wedges are identified that contain only the unqiue negative eigenvalue of A. These results lead to new eigenvalue location regions for nonnegative matrices.