The Properties And Applications Of Several Generalized Circulant Matrices

Circulant matrix class has become a very active and important research direction in matrix theory and applied mathematics.It is also widely applied to the modern science and technology.Deep study of cyclic matrices advanced the knowledge of matrix theory and provided a better mathematical calculation method for the development of other disciplines.It is a very meaningful work.The main contents and conclusions of this dissertation are shown in detail below:1.This dissertation studies two types of quasi-cyclic matrix for the first time,the quasi-1-circulant matrix and quasi-m-circulant matrix.Based on the special form of those two types of quasi-cyclic matrix,this study gets the sufficient and necessary conditions for determining a matrix to be the quasi-circulant matrix.Then,the non-singularity,decomposition and the seal of the operation of the quasi-circulant matrix are discussed as well.2.According to the special nature of the quasi-circulant matrix and the opti-mization theory,we discuss the least-squares problem of matrix equations AX = B,XC = D for the quasi-circulant matrix.The explicit representations of the general solution can be derived.Finally,this paper consider the inverse eigenvalue problem of the quasi-circulant matrix.3.The least-squares problenms and the inverse eigenvalue problem for Hankel-circulant matrix and Hankel-skew-circulant matrix are discussed.Combining the op-timization theory and the properties of circulant matrices,the least-squares problem can be transformed into a simple problem of linear equation Ty = b.Therefore,the explicit representations of the general solution can be derived.Moreover,the coefficient matrix Q of the linear equation related to the matrix X with constraint conditions is a circulant matrix.Thus,the explicit representations of the general and unique solution are derived.Besides,the determination conditions of the unique solution are obtained by the generalized 1 norm.Finally,necessary and sufficient conditions under which the problem is solvable are presented,and necessary and sufficient conditions under which the problem is uniquely solvable are also discussed.