文档介绍：§ 相似矩阵一、相似矩阵的概念二、相似矩阵的性质三、 n 阶矩阵与对角矩阵相似的充要条件 Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. 一、相似矩阵的概念定义 1设A,B为n阶方阵,如果存在可逆矩阵 P,使得 P -1AP=B 成立,则称矩阵 A与B相似,记为 A?B。称 P为相似变换矩阵。相似关系是矩阵间的一种等价关系,即满足自反性: A ?A , 对称性: 若A?B,则B ?A 传递性:若 A?B, B ?C, 则A ??C Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. 1. 如果方阵 A与B相似,则它们有相同的特征多项式,从而有相同的特征值。即若A?B,则|?E-A |=|?E-B| |?E-B| =|P -1(?E)P -P -1 AP | =|?E-P -1 AP | =|P -1(?E-A)P| =|P -1|?|?E-A|?|P| =|?E-A|, 二、相似矩阵的性质 A与B有相同的特征多项式, 所以它们有相同的特征值。 2. 相似矩阵的行列式相等。即若 A~B ,则|A|=|B| |B| =|P -1 AP|=|P -1 | |A| |P|=|A| |P -1 P| =|A| 证明: 因为 P -1 AP =B, Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. 。即若 A~B ,则 1 1 1 n n n i i i i i i i i a b ?? ??? ?? ?? ,或者都不可逆。若都可逆,其逆矩阵也相似。??.,.3 为正整数相似与则相似与若mBABA mm Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. 5. 相似矩阵有相同的秩。即若 A~B ,则 R(A) =R (B) 注意:以上性质均为相似的必要条件,可以用来排除哪些矩阵不相似。的所有的特征值。是, 相似,则与对角阵、若,A An n???????? 21 2 16?????????????? Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pt