Morris Newman

A note on cospectral graphs

Abstract The notion of a (1, x ) adjacency matrix is introduced, together with methods for dealing with it. It is shown that in many instances this adjacency matrix is superior to the usual (0, 1) adjacency matrix, and will distinguish cospectral pairs, when the latter will not. In those cases in which the (1, x ) adjacency matrix fares no better than the (0, 1) adjacency matrix, a good deal can be said about the matrices by which the cospectral pairs are similar.

Positive definite matrices and Catalan numbers

It is shown that the number of n X n integral triple diagonal matrices which are unimodular, positive definite and whose sub and super diagonal elements are all one, is the Catalan number (M)/(n + 1). More generally, it is shown that if A is a fixed integral symmetric matrix and d is a fixed positive integer, then there are only finitely many integral diagonal matrices D such that A + D is positive definite and det(A + D) = d.

Invariant factors of combinatorial matrices

The Smith normal forms of an Hadamard matrix of order 4m (m square-free), and of the incidence matrix of a (ν, k, λ) configuration (n=k−λ square-free (n, λ)=1), are determined.

Counting modular matrices with specified Euclidean norm

Let N(x) denote the number of matrices A = a bc d belonging to SL(2, Z) such that a2 + b2 + c2 + d2 ⩽ x. It is shown that the limit as x approaches infinity of N(x)x equals 6.

On a theorem of Čebotarev

A new proof is given of the theorem that no submatrix of the p×p matrix S=(ζ(i−1)(j−1)) is singular, where ζis a primitive pth root of unity and p is a prime. Some related results are also discussed.

Multiplicativity of the Smith Normal Form

New conditions are derived which are sufficient to make the Smith Normal Form a Multiplicative matrix function.

A result on integral symmetric matrices

Let A be an integral matrix such that det A = 1 mod m A ≡ AT mod m, where m is odd. It is shown that a symmetric integral matrix B of determinant 1 exists such that B ≡ A mod m. The result is false if m is even.

Continuous solutions of a homogeneous functional equation

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A note on matrix equivalence

Let R be a principal ideal ringRn the ring of n × n it matrices over R. It is shown that if A, B, X, Y are elements of R* such that A = XB, B = YA, then A and B are left equivalent. Some consequences are given.

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AAT is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.