关于复对称矩阵的一些性质的讨论

2014-04-24 01:15

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.

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

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