Let H be any complex inner product space with inner product < ·, · >. We say that f : | C → | C is Hermitian positive definite on H if the matrix ( f(< z,z >) )n r,s=1 (∗) is Hermitian positive definite for all choice of z, . . . ,z in H, all n. It is strictly Hermitian positive definite if the matrix (∗) is also non-singular for any choice of distinct z, . . . ,z in H. In this article we prove...

We give a simple example to show that a result on the equivalent singular points of systems of ordinary linear differential equations due to G. D. Birkhoff [3; 5, pp. 252-257] needs amendment. In matrix notation, in which Y, P, etc. denote nXn matrix-valued functions of a complex variable z, this result is as follows. A. Result. Every linear differential system (1) Y'(z) = P(z)Y(z) with a singu...

We present an algorithm for the computation of a shifted Popov Normal Form of a rectangular polynomial matrix. For specific input shifts, we obtain methods for computing the matrix greatest common divisor of two matrix polynomials (in normal form) or such polynomial normal form computation as the classical Popov form and the Hermite Normal Form. The method is done by embedding the problem of co...

We considered the ordered components, Y , of a multivariate random variable, Y, with covariance matrix Σ11 and the vector, Z , of concomitantly or induced ordered components of a secondary random vector, Z, with covariance matrix Σ22. Assuming that Σ11, Σ22 and the covariance structure between Y and Z are permutation-symmetric, the joint covariance structure for Y and Z is obtained. The case in...

We consider chip-firing dynamics defined by arbitrary M-matrices. M-matrices generalize graph Laplacians and were shown by Gabrielov to yield avalanche finite systems. Building on the work of Baker and Shokrieh, we extend the concept of energy minimizing chip configurations. Given an M-matrix, we show that there exists a unique energy minimizing configuration in each equivalence class defined b...

where Y, Z, Kix) and G(x) are « x n matrices and each of Kix) and G(x) is a symmetric matrix of continuous functions on a z% x < oo. By a solution of (b) we mean a pair of n x n matrices {T(x),Z(x)} such that each of the elements of Y and Z is a differentiable function and such that {Y,Z} satisfies the initial condition and satisfies the matrix differential system almost everywhere on a S! x < ...

On a closed Riemann surface Ra of genus g there exist g linearly independent differentials of the first kind, wu • • • , ws,.and their integrals around 2g canonical cycles or retrosections, Oi, • • • , a„, bi, ••• , ba, are usually put together to form a gX2g matrix (an; ßn) = (A; B), i, j =1, • • • , g, where a,-,is the integral of w{ over a,. Riemann showed that the entries in this matrix wer...

We obtain an asymptotic upper bound for the minimal number of generators for a finite direct sum of matrix algebras with entries in a finite field. This produces an upper bound for a similar quantity for integer matrix rings. We obtain an exact formula for the minimal number of generators for a finite direct sum of 2-by-2 matrix algebras with entries in a finite field. As a consequence, we show...

The aim of the present paper is to analyze the behavior of Fiedler companion matrices in the polynomial root-finding problem from the point of view of conditioning of eigenvalues. More precisely, we compare: (a) the condition number of a given root λ of a monic polynomial p(z) with the condition number of λ as an eigenvalue of any Fiedler matrix of p(z), (b) the condition number of λ as an eige...

in this paper, the maximal dissipative extensions of a symmetric singular 1d discrete hamiltonian operator with maximal deficiency indices (2,2) (in limit-circle cases at ±∞) and acting in the hilbert space ℓ_{ω}²(z;c²) (z:={0,±1,±2,...}) are considered. we consider two classes dissipative operators with separated boundary conditions both at -∞ and ∞. for each of these cases we establish a self...