Research On Some Problems Of Infinite Companion Matrix

Following the progression of engineering technology during 20th century, matrix theory is developed as a mathematical method. At the same time, the invention of computer drives the compute mathematics further to apply in more fields. Today, matrix theory has been extensive application as a basic tooling used for mathematics study. As an output of engineering calculation, matrix calculation can be applied in lots of fields. For example, the optical spectrum analysis of substance may be use singular value & spectrum theory of matrix. Perturbation theory of matrix is often used to error analysis in large amount of data. The inherent property study of normal matrices has an excellent guide meaning for us. Otherwise, it is equaled important for us to study the special matrices. Not only for that, these special matrices are used as important elements for us to study the whole matrices group. They just look like 0 & 1 onto all natural numbers.This paper is major to discuss some special matrices, like the companion matrix and infinite companion matrix. Especially to infinite companion matrix, Vlastimil Ptak have a deeply study about it and also developed the concept of unidirectional infinite companion matrix. Due to his work esprit, arouse my interest into the study of companion matrix.This paper total includes four sections as below.The first section is the introduction of matrix background and some idea of mine.The second section mainly introduces the concept of normal companion matrices and their property. The study of companion matrix is developed by a characteristic polynomial. If matrix A is well known, its characteristic polynomial onto companion matrix has the same eigenvalues with the matrix A. This property of finite companion matrix makes it has close ties with (characteristic) polynomial, noted the relations between (eigenvalues) root and polynomial's coefficient. Taking this result into other matrices and polynomial, combining companion matrices, we can get the nature of relationship along some special matrices.The third section has expanded the accomplishment of Vlastimil Ptak's study. Systematically expounded the concept and property of infinite companion matrix from a polynomial p(z). Vlastimil Ptak found that, ifw(z)= z~n, then a pair polynomial (p(z), w(z)) of degree n onto infinite companion matrix is a special circumstances. It related to Bezoutians, projection operators, reproducing kernels, and dilation theory.In the last part, it major summarized some conclusions from all above, and illustrated with example to show the application of infinite companion matrix.General of all, we introduce the generating function in special space and infinite companion matrix from polynomial and normal companion matrix, and then combine theory and its application.