The distribution of eigenvalues and the upper bounds for spread interval matrices are significant in various fields mathematics applied sciences, including linear algebra, numerical analysis, control theory, combinatorial optimization. We present within determine their using Geršgorin’s theorem. Specifically, through an equality variance a discrete random variable, we derive symmetric matrices....

We examine the problem of equivalence of discrete time auto-regressive representations (DTARRs) over a finite time interval. Two DTARRs are defined as fundamentally equivalent (FE) over a finite time interval [0, N ] if their solution spaces or behaviours are isomorphic. We generalise the concept of strict equivalence (SE) of matrix pencils to the case of general polynomial matrices and in turn...

In the first part of these notes, we review some of the recent developments in the study of the spectral properties of Wigner matrices. In the second part, we present a new proof of a Wegner estimate for the eigenvalues of a large class of Wigner matrices. The Wegner estimate gives an upper bound for the probability to find an eigenvalue in an interval I, proportional to the size |I| of the int...

We determine the exact range of the smallest and largest eigenvalues of real symmetric matrices of a given order whose entries are in a given interval. The maximizing and minimizing matrices are specified. We also consider the maximal spread of such matrices.