Canonical Forms And Numerical Methods For The Eigenvalue Problems Of Some Doubly Structured Matrices

Structured matrices occur in many scientific and engineering applications, and structured matrix eigenvalue problems are widely used. In this paper, Jordan canonical forms for 12 kinds of doubly structured matrices are discussed. It is shown that the eigenvalues of 6 kinds of doubly structured matrices are special given numbers. As applications, 3 open questions raised by Tisseur(2003) concerning structured backward error are solved. What' s more, structure-preserving algorithms for the eigenvalue problems of 4 kinds of doubly structured matrices are developed, in which algorithms using wherever possibly only real arithmetics are developed for 2 kinds of doubly structured matrices.This paper is composed of 4 sections.In section 1, the historic development of investigation concerning structured matrix eigenvalue problems is summarized. Main progress at present is outlined, and canonical form problems for structured eigenvalue problems which are practically significant but have not been investigated are listed. Furthermore, the purpose and main results of this paper are briefly shown.In section 2, Jordan canonical forms for 12 kinds of doubly structured matrices are investigated. The results are new.In section 3, structure-preserving algorithms for 12 kinds of doubly structured matrices under consideration are developed by the exploitation of the newly derived Jordan canonical forms. This section consists of 4 subsections. In the first subsection, it is shown that the eigenvalues of the first 6 kinds of doubly structured matrices are already given and 3 open questions raised by Tisseur(2003) concerning structrued backward error are solved.In the second subsection, structure-preserving algorithms using wherever pos-sibly only real arithmetics are developed for the seventh kind and the eighth kind of doubly structured matrices, respectively.In the third subsection, structure-preserving algorithms for the nineth kind and the tenth kind of doubly structured matrices are derived by using comlpex simplectic transformations.In the fourth subsection, a numerical example is given for the eigenvalue problem of the tenth kind of doubly structured matrix to test the newly developed structure-preserving algorithms.Finally, in section 4, some remarks on the main results in this paper are given.