In this paper, the inverse eigenvalue problem for the bordered diagonal matrices are reconsidered whose elements are equal to zero except for the first row, the first column and the diagonal elements. The necessary and sufficient conditions for existence of a symmetric bordered diagonal matrix from special spectral data have been determined. A new algorithm to make such matrices is derived and some numerical examples are given to illustrate the efficiency of the method.

For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties. We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous differentiability. eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.

We investigated maximal Prym varieties on  nite  elds by attaining their upper bounds on the number of rational points. This concept gave us a motivation for de ning a generalized de ni-tion of maximal curves i. e., maximal morphisms. By MAGMA, we give some non-trivial examples of maximal morphisms that results in non-trivial examples of maximal Prym varieties.

Smart systems in general, and specifically neural networks, are expected to be of great assistance in large scale matrix computation. However it was necessary to work with smaller problems at first step. To this end, a method based on using perceptrons and Kohonen networks for determining the first three modal frequencies of frames of up to 20 stories high and two bays wide from their stiffness and mass matrices, was developed. This paper introduces this modular network as a preferable alternative for designing smart systems for learning large scale mapping problems in structural engineering, and in addition, reports the successful application of this modular network to the eigenvalue and modal frequency determination of shear frame structures.

Let $\mathtt{A}(\mathtt{G})$ be the adjacency matrix of a simple connected undirected graph $\mathtt{G}$. A graph $\mathtt{G}$ of order $n$ is said to be non-singular (respectively singular) if $\mathtt{A}(\mathtt{G})$ is non-singular (respectively singular). The spectrum of a graph $\mathtt{G}$ is the set of all its eigenvalues denoted by $spec(\mathtt{G})$. The anti-reciprocal (respectively reciprocal) eigenvalue property for a graph $\mathtt{G}$ can be defined as `` Let $\mathtt{G}$ be a non-singular graph $\mathtt{G}$ if the negative reciprocal (respectively positive reciprocal) of each eigenvalue is likewise an eigenvalue of $\mathtt{G}$, then $\mathtt{G}$ has anti-reciprocal (respectively reciprocal) eigenvalue property ." Furthermore, a graph $\mathtt{G}$ is said to have strong anti-reciprocal eigenvalue property (resp. strong reciprocal eigenvalue property) if the eigenvalues and their negative (resp. positive) reciprocals are of same multiplicities. In this article, graphs satisfying anti-reciprocal eigenvalue (or property $(-\mathtt{R})$) and strong anti-reciprocal eigenvalue property (or property $(-\mathtt{SR})$) are discussed.

This paper considers an inverse eigenvalue problem for bisymmetric nonnegative matrices. We first discuss the specified structure of the bisymmetric matrices. Then for a given set of real numbers of order maximum five with special conditions, we construct a nonnegative bisymmetric matrix such that the given set is its spectrum. Finally, we solve the problem for arbitrary order n in the special case of the spectrum.