Circulant Matrix & Circulant Matrix Equation And Its Stability Analysis

Circulant matrix is a kind of very important special matrix. It has extensive application in the modern science and technology engineering. In this paper, we introduce the definition of the circulant matrix and its basic properties. And further more we get the concept of generalized circulant matrix and its basic properties. Using the special properties of the circulant matrix we simplify the circulant matrix calculation method about the determinant, eigenvalue, eigenvector and inverse. In this paper we also discuss the generalized circulant matrix and get the existence of the solution, and study the structure of the solution. Using the special structure of the solution, we give the general method of computing the inverse of the generalized circulant matrix. Finally, we discuss the stability of the solution of the circulant linear equation set and get the formulas and estimations about the condition of the solution. This paper is mainly composed of six chapters:In the first chapter, we introduce the background knowledge about the theory system, including its development history, the previous research, and its application value in technology production.In the second chapter, we generalize the property of the circulant matrix and get several good properties about circulant block matrices and partitioning circulant matrix.In the third chapter, we summarize some of the calculation on circulant matrix, and give the necessary and sufficient conditions of the invertibility of the circulant matrix and give two classic algorithms for computing the inverse.In the fourth chapter, we analyze the existence of solutions about the generalized circulant matrix equation, and get the structure of the solution when there is a solution. Further, we get the method of getting the inverse of circulant block matrices and partitioning circulant matrix..In the fifth chapter, we discusses the stability of the solution about generalized circulant linear equation set and get the expression of condition number and estimates the upper and lower bounds.