Research On The Moore-Penrose Inverse And The Weighted Drazin Inverse Of Block Circulant Matrices And Block K-circulant Matrices

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 circulant matrix, block circulant matrices and block k-circulant matrices. Especially to block circulant matrix, R L Smith have a deeply study about it and also developed the concept of block circulant matrix. Due to his work esprit, arouse my interest into the study of block circulant 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 introduce the concept of normal circulant matrix and their property. The inverse of Moore-Penrose in circulate matrix is discussed, and it based on the diagonalization of circulate matrix. We illustrated that with examples. The way above is simple to discuss the inverse for we only need the properties of inverse of Moore-Penrose, can constrate the form of inverse of Moore-Penrose.In the third part we will extend some results about circulate matrix. The concept of block circulant matrix is discribed as well as some properties and on this basis we introduced the solution of the inverse of Moore-Penrose and the inverse of Drazin with weight. The calculations are simple to given the diagonalization of circulate matrix according the second part in the paper.The forth part is the generation of the third part. We extend the block circulate matrix to block k-circulate matrix. Accordingly, the inverse of k-circulate matrix and the inverse of Drazin with weight are presented which we using the properties of diagonalization of circulate matrix in the third part. The inverse of Moore-Penrose of block k-circulate matrix is discussed in some works, but these results all associated with unit mode. In this paper the results for k∈are right, so generate the previous work.Overall, this paper determined the form of diagonalization, given the inverse of Moore-Penrose and Drazin with weight through calculation, combined with examples.