Algunas propiedades de las matrices de Toeplitz complejas

Abstract

This monograph is using as a reference The chapter four of the article “On some properties of Toeplitz matrices ”, this article emphasizes Toeplitz matrices over the field of complex numbers. The objective of this monograph is to make an indepth study of the results given in the above mentioned article, for this I will show the step-by-step prove of each result, exemplifying the main results and giving some applications of these matrices. Mainly I made a study of the Frobenius and Operator norms defined over the subalgebra of Toeplitz matrices. One of the main results gives a relationship between the set M_n(C) and its respective subalgebra of Toeplitz matrices, this relationship can lead to the possible extension of the applications shown at the end of this monograph. Lin’s theorem has great importance since it is necessary to prove the aforementioned theorem. Another important result is the theorem 5.5 since it allows generalization to the area of C^{*} −algebras . Finally, one of the objectives in this monograph is to relate the circular matrices to the Discrete Fourier Transform (DFT), this relationship is given in chapter five with a study of the main properties of the DFT.