| In this paper, we mainly analyse the problem of determinant, explicit inverse and the structure perturbation analysis of five kinds of structure matrix (skew circulant matrices, BCSCB matrix, BSCCB matrix, row first-plus-last Toeplitz matrix and row first-plus-last Hankel matrix), and the main contents of the thesis consist of six chapters:The first chapter includes three parts, in the first part, we introduce applied background and the current research situation of skew circulant matrix, circulant matrix, and Toeplitz matrix in the domestic and overseas. In the second part, we give the definitions of five kinds of structure matrices and important lemmas, which are closely related to the following research. In the third part, we simply introduce main work of this paper.The second chapter, we consider the fast algorithm for the determinant of k-diagonal skew circulant matrix and the computational complexity. There are three parts in this chap-ter. In the first part, we propose the fast algorithm for the determinant of k-diagonal skew circulant matrix, and when the dimension tends to infinity, an algorithm is also provided. In the next part, we consider the computational complexity about the algorithm mentioned above. Finally, the algorithm of computing the determinant on symmetric skew circulant matrices with integer entries is provided.In chapter Three and Four, we analyse the structure perturbation about the linear system with the coefficient matrices are BCSCB matrix and BSCCB matrix, respectively. Based on the style spectral decomposition of basic skew circulant and circulant matrices, the block style spectral decomposition of BCSCB and BSCCB matrices are obtained. Next, we consider the structure perturbation analysis, include the condition number, relative error and the lower bound of the optimal backward perturbation. Simultaneously, algorithms are provided to obtain the lower bound. Numerical examples are provided to verify the effectiveness of the algorithms.In the fifth chapter, we focus on the explicit inverses of row first-plus-last Toepitz matrix (TRFPL) and row first-plus-last Hankel matrix (HRFPL), and then the stability of the inverse formulae of TRFPL and HRFPL matrices are discussed. Finally, examples are provided to verify the feasibility of the algorithms.In the last chapter, we summarize the main work of the paper, then give some sugges-tions and expectations to the research in the future. |
