Let F be a strongly regular graph with adjacency matrix A. Let I be the identity matrix, and J the all-1 matrix. Let p be a prime. Our aim is to study the p-rank (that is, the rank over Fp, the finite field with p elements) of the matrices M = aA + bJ + cI for integral a, b, c. This note is based on van Eijl [8]. 1. The Smith normal form Let us write M ~ N for integral matrices M and N of order...

The remarks of this note are concerned with a result on transformations stated below as Theorem A, and are two-fold in nature: firstly, there are comments on the relation of this theorem to results of Perron [3] and Diliberto [l; 2], in the hope of correcting a misunderstanding that has arisen in this regard; secondly, there are remarks stressing two general properties of admissible transformat...

where Idn is the n×n identity matrix in R, andA = (aij). We can’t quite invert this matrix, but we almost can. Recall that any n×nmatrixM has an adjoint matrix adj(M) such that adj(M)M = det(M)Idn. The coefficients of adj(M) are polynomials in the coefficients of M. (You’ve likely seen this in the form of a formula for M when there is an inverse.) Multiplying both sides of (1) on the left by ad...

We consider the algebraic Riccati equation for which the four coefficient matrices form an M -matrix K. When K is a nonsingular M -matrix or an irreducible singular M -matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M ...

We present a test for determining if a substochastic matrix is convergent. By establishing a duality between weakly chained diagonally dominant (w.c.d.d.) Lmatrices and convergent substochastic matrices, we show that this test can be trivially extended to determine whether a weakly diagonally dominant (w.d.d.) matrix is a nonsingular M-matrix. The test’s runtime is linear in the order of the in...

Let A = M N E IR nn be a splitting. We investigate the spectral properties of the iteration matrix M 1 N by considering the relationships of the graphs of A, M, N, and M1 N. We call a splitting an M-splitting if M is a nonsingular M-matrix and N;;. O. For an M-splitting of an irreducible Z-matrix A we prove that the circuit index of M1 N is the greatest common divisor of certain sets of integer...

The n×m matrix AT obtained by exchanging rows and columns of A is called the transpose of A. A matrix A is said to be symmetric if A = AT . The sum of two matrices of equal size is the matrix of the entry-by-entry sums, and the scalar product of a real number a and an m× n matrix A is the m× n matrix of all the entries of A, each multiplied by a. The difference of two matrices of equal size A a...

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n×m matrix M into a product of a nonnegative n × d matrix W and a nonnegative d ×m matrix H. A longstanding open question, posed by Cohen and Rothblum in 1993, is whether a rational matrix M always has an NMF of minimal inner dimension d whose factors W and H are also rational. We answer this question negat...

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n×m matrix M into a product of a nonnegative n× d matrix W and a nonnegative d ×m matrix H . A longstanding open question, posed by Cohen and Rothblum in 1993, is whether a rational matrix M always has an NMF of minimal inner dimension d whose factors W and H are also rational. We answer this question negat...

Let F be a k×` (0,1)-matrix. We say a (0,1)-matrix A has F as a configuration if there is a submatrix of A which is a row and column permutation of F . In the language of sets, a configuration is a trace and in the language of hypergraphs a configuration is a subhypergraph. Let F be a given k × ` (0,1)-matrix. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. The...