When R is a commutative ring, matrices A and B in Mn(R) are called conjugate when UAU−1 = B for some U ∈ GLn(R). The conjugacy problem in Mn(R) is: decide when two matrices in Mn(R) are conjugate. We want to look at the conjugacy problem in Mn(Z), where ideal theory and class groups make an interesting appearance. The most basic invariant for conjugacy classes of matrices is the characteristic ...

For matrix power series with coefficients over a field, the notion of a matrix power series remainder sequence and its corresponding cofactor sequence are introduced and developed. An algor!thm for constructing these sequences is presented. It is shown that the cofactor sequence yields directly a sequence of Padd fractions for a matrix power series represented as a quotient B(z)-lA(z). When B(z...

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...

all 1’s. But a matrix of this form with order k, z # 0, and JOHN A. HOROS AND MARTIN E. HELLMAN yk + z # 0, has for its inverse the matrix (y/z(yk + z))U + (l/z)Z. It follows that the inverse of the coefficient matrix is Abstract-A confidence model for finite-memory learning systems is advanced in this correspondence. The primary difference between this and the previously used probability-of-er...

Let M be an n × n square matrix and let p(λ) be a monic polynomial of degree n. Let Z be a set of n × n matrices. The multiplicative inverse eigenvalue problem asks for the construction of a matrix Z ∈ Z such that the product matrix MZ has characteristic polynomial p(λ). In this paper we provide new necessary and sufficient conditions when Z is an affine variety over an algebraically closed field.

sgn(z) = ( 1 if Re(z) > 0, −1 if Re(z) < 0 for z ∈ C lying off the imaginary axis. Next, the matrix sign function is given by sgn(M) = U sgn(Λ)U−1 where M ∈ C n×n is a Hermitian matrix with no eigenvalues on the imaginary axis. The unitary n×n matrix U has column vectors equal to the orthonormal eigenvector basis of C n determined byM , and the diagonal n×n matrix Λ has components equal to the ...

In this paper, we discuss stability and linear independence of the integer translates of a scaling vector = (1 ; ; r) T , which satisses a matrix reenement equation (x) = n X k=0 P k (2x ? k); where (P k) is a nite matrix sequence. We call P (z) = 1 2 P P k z k the symbol of. Stable scaling vectors often serve as generators of multiresolution analyses (MRA) and therefore play an important role ...

We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large N matrix model. Using a related function Ξ(z) we develop an analog between this function and the Airy function Ai(z) of the Gaussian matrix model. The analog...

For a given pair of s-dimensional real Laurent polynomials (~a(z),~b(z)), which has a certain type of symmetry and satisfies the dual condition~b(z) T ~a(z) = 1, an s× s Laurent polynomial matrix A(z) (together with its inverse A−1(z)) is called a symmetric Laurent polynomial matrix extension of the dual pair (~a(z),~b(z)) if A(z) has similar symmetry, the inverse A−1(Z) also is a Laurent polyn...

Let g(z, x) denote the diagonal Green's matrix of a self-adjoint m × m matrix-valued Schrödinger operator H = − d 2 dx 2 Im + Q(x) in L 2 (R) m , m ∈ N. One of the principal results proven in this paper states that for a fixed x 0 ∈ R and all z ∈ C + , g(z, x 0) and g ′ (z, x 0) uniquely determine the matrix-valued m × m potential Q(x) for a.e. x ∈ R. We also prove the following local version o...