关于复对称矩阵的一些性质的讨论

关于复对称矩阵的1些性质的讨论

摘要：本论文分析和阐述了复对称矩阵在相关方面的性质和问题，包括特征值、条件数和对角化等问题．先给出了1些常用到的复对称矩阵的结论，这对后面展开的研究与分析很有用．再分别给出了复对称矩阵的特征值和条件数．复对称矩阵特征值问题则突破了常用的在实矩阵中的计算方法，将复对称矩阵特征值分成了实部矩阵的特征值和虚部矩阵的特征值两个部分来研究，从而简化了运算．另外，本文找出了复对称矩阵与1般的复矩阵的关系，即每个复矩阵都相似1个复对称矩阵．在研究复对称矩阵的合同和相似时，对复对称矩阵和若当形矩阵重新作了归纳和总结．在论文的结尾，本文还对复矩阵中常用的Schur定理作了简要介绍和证明．

关键词：复对称矩阵; 特征值; 条件数; 若当形矩阵

Argument of Characterizations of Complex Symmetric Matrices

Abstract: In this paper, the problems of correlation characterizations of complex symmetric matrices are deduced and analyzed, which include the eigenvalue, the condition number and the problem of diagonalization. Begin with analyzing structure of complex symmetric matrices, the author summarizes some usually used conclusions to study and analyze what follows in the passage. And then he respectively introduces the eigenvalues and the condition number of complex symmetric matrices. The problem of eigenvalues of complex symmetric matrices is discussed by an unusual method which does not apply to that of a real matrix. This method will separate eigenvalues of complex symmetric matrices to two parts: that of a real matrix and that of a complex matrix. So it simplifies operations of arithmetic. In addition, the connection of complex matrices and complex symmetric matrices , namely every complex matrix is similar to a complex symmetric matrix, is found. Especially when the author thinks deeply the problem of diagonalization of complex symmetric matrices, he makes much work for congruence and similarity of complex symmetric matrices and Jordan matrices and draws many new conclusions. In the end of this paper, the theorem of Schur is introduced simply and fully proved, which is used to the study of a complex matrix.

Keywords: complex symmetric matrices; eigenvalue; condition number; Jordan