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.

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.

The undirected power graph of a  nite 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  of A is a Main eigenvalue if the eigenspace "( ) has an eigenvector X such that Xtj 6= 0, where j is the all-one vector. In this paper we want to focus on the power graph of the  nite cyclic group Zn and  nd a condition on n where P(Zn) has exactly one Main eigenvalue. Then we calculate the number of Main eigenvalues of P(Zn) where n has a unique prime decomposition n = prp2. We also formulate a conjecture on the number of the Main eigenvalues of P(Zn) for an arbitrary positive integer n.

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.