11.7特征值,对角化.ppt

1§ 矩阵的特征值化H矩阵为对角形矩阵特征值与特征向量化H矩阵为对角形 Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. 2 工程中的一些问题,如振动问题和稳定性问题,常可归结为求一个方阵的特征值和特征向量的问题;而数学中诸如方阵的对角化,求线性变换的不变元素等问题也需要特征值和特征向量的概念. 而矩阵的对角化涉及到如何把一个二次型化成对角形,进一步化成标准形的问题,解析几何中的提法是:对二次曲线和二次曲面的一般方程通过一个坐标变换化成标准方程. Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. 3 说明., 言的特征值问题是对方阵而特征向量?x??.0 ,0 ,.2 的特征值都是矩阵的即满足方程值有非零解的就是使齐次线性方程组的特征值阶方阵 A EA xEA An????????一、特征值与特征向量. ,,, ,1 的特征向量的对应于特征值称为量非零向的特征值称为方阵这样的数那末成立使关系式维非零列向量和如果数阶矩阵是设定义????Ax A x Ax xnnA?而且若 x是特征向量,则乘以非零常数后仍是. Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Evaluation only. Created with Client Profile . Created with Client Profile . Copyright 2004-2011 Aspose Pty Ltd. Copyright 2004-2011 Aspose Pty Ltd. ??EA??0 21 222 21 112 11??????? nn nn n naaa aaa aaa???????次方程为未知数的一元称以 n? 0??EA?.的为A 特征方程,, 次多项式的它是 n?记?? EAf???? A 特征多项式 按代数基本定理,知对 n阶方阵恰有 n个特征值(重根按重数计).特征值有时也