Is every matrix similar to a Toeplitz matrix?

Abstract

[en] The theory of matrices is a very important field in the world of mathematics, which plays a central role in the study of a large variety of areas in pure and applied mathematics. Is is one of the first topics studied in the degree. I remember when I was studying the first course that I got fascinated by the fact that you can transform a matrix into an easier one through a similarity and these two matrices would preserve basic properties that give a big amount of information of the matrix. In Algebra courses, we have learned all the geometrical concepts that a matrix can represent and are endless. Besides, there are structured matrices, such as Toeplitz matrices which are useful in many branchs. Their study has been an active field of research since the beginning of the last century, and it remains until today. Many journal papers have been devoted to these matrices and the large interest has a reason: the number of applica- tions that arise from it. They contribute to the discretization of differential and integral equations, in the theory of orthogonal polynomials, trigonometric moments, time series analysis, graph theory and signal processing applications, as it is studied at [2], [9], [14] and [18]. The aim of this work is to bring together these two areas. We are interested in finding which Jordan canonical forms can be realized by some Toeplitz matrix. In other words, if every matrix is similar to a Toeplitz matrix. To do so, we have deeply studied the two only articles published related to our topic ([4] and [5]). Before starting with this research, we will present all concepts that will be needed related to Linear Algebra and Toeplitz matrices. Once we have set the knowledge base required, we will focus on our final purpose. Unless it is otherwise stated, we suppose we are in the field of complex numbers. With a view to answer our question, we have divided the study in different parts, mainly in two blocks. The first block is dedicated to proving those matrices which are similar to a Toeplitz matrix, which are diagonalizable matrices, nonderogatory matrices and the general case for $n \leq 4$. Regarding the second part, is devoted to matrices of general dimension and we will need to do some strong assumptions like having just one eigenvalue. In this case, we distinguish between matrices of odd and even order. All these investigations will lead us to answer this question.