华南农大高数第8章相似矩阵与二次型2.ppt

Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. ●相似矩阵设 A和B为 n 阶矩阵。如果存在 n 阶可逆矩阵 P, 使得,则称 A相似于 B,或说 A和B相似。 1 P AP B ??基本性质(1)反身性 A 相似于 A。(2)对称性 A 相似于 B,则 B相似于 A。(3)传递性 A 相似于 B,B相似于 C,则 A相似于 C。定义 Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. ●相似矩阵的性质 1 P AP B ?? 1 P AP B ?? ? 1 P A P B ?? ? 1 1 P P A P P A A B ? ???? ?若A和B相似,则( 1) (2) ( ) ( ) R A R B ? A B ?证明( 1) 1 P AP B ?? 1 ( ) ( ) R P AP R B ?? ?( ) ( ) R A R B ? ?(2) 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 P AP B ??? 1 ( ) ( ) tr B tr P AP ?? 1 ( ) ( ) tr APP tr A ?? ? 1 P AP B ?? B E ?? 1 P AP E ??? ? 1 ( ) P A E P ??? ? 1 P A E P ??? ? A E ?? ?(3) ( ) ( ) tr A tr B ?(4) A E B E ? ?? ??证明证明(特征多项式相同) (有相等的迹) Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. 推论如果 n阶方阵 A相似于对角形矩阵 12n???? ?? ?? ???? ?? ?? ?? ??则是A的全部特征值。 1 2 , , , n ? ? ?? Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose P