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<TITLE> 特徵值与特徵向量 </TITLE>
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<H1>5.3.2 特徵值与特徵向量</H1>
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假设<FONT FACE="Times New Roman"> A</FONT>为一个<FONT FACE="Times New Roman">
<IMG SRC="img00004-2.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00004.gif"> </FONT>矩阵，而<FONT FACE="Times New Roman">
X </FONT>为一个有<I><FONT FACE="Times New Roman">n</FONT></I>列的栏向量，<IMG SRC="img00005-2.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00005.gif">为一纯量。考虑以下的数学式
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<IMG SRC="img00006-2.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00006.gif">
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如果<FONT FACE="Times New Roman">X</FONT>由不为零的元素所组成，其中<IMG SRC="img00007-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00007.gif">要满足上式称为矩阵<FONT FACE="Times New Roman">A</FONT>的特徵值<FONT FACE="Times New Roman">(eigenvalue)</FONT>，而<FONT FACE="Times New Roman">X</FONT>称为矩阵<FONT FACE="Times New Roman">A</FONT>的特徵向量
<FONT FACE="Times New Roman">(eigenvector)</FONT>。特徵向量代表一个正规正交<FONT FACE="Times New Roman">(orthonormal) </FONT>的向量组，所谓的正规正交向量，是指这向量与自身做
内积的值为一单位向量；在几何关系上是指二量相互垂直且此其内积值再做正规化<FONT FACE="Times New Roman">(normalization)</FONT>。<BR>
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上式也可改写为
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<IMG SRC="img00008-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00008.gif">
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其中<FONT FACE="Times New Roman"> I </FONT>为<IMG SRC="img00009-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00009.gif">单位矩阵。
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<IMG SRC="img00010-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00010.gif">
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则eigenvalue可以用特徵方程式计算
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<IMG SRC="img00012-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00012.gif">
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上述的二次方程式可求解二个根分别为<IMG SRC="img00013-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00013.gif">，这二个值即为<FONT FACE="Times New Roman">A</FONT>的特徵值。而<FONT FACE="Times New Roman">A</FONT>的特徵向量求法如下
，分别将任一特徵值代入<IMG SRC="img00014-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00014.gif">。例如<IMG SRC="img00015-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00015.gif">
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<IMG SRC="img00016-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00016.gif">
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另一个特徵值<IMG SRC="img00017-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00017.gif">代入，可以得到另一个特徵向量为<IMG SRC="img00018-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00018.gif">。我们可找到无限多个向量，满足上述的
特徵向量，例如
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<IMG SRC="img00019-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00019.gif"><BR>
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因此要得到唯一的特徵向量，即是正交<FONT FACE="Times New Roman">(orthonormal)</FONT>特徵向量组<FONT FACE="Times New Roman">Q</FONT>，利用其特性<IMG SRC="img00020-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00020.gif">
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<IMG SRC="img00021-1.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00021.gif">
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求解上式可得<IMG SRC="img00022.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00022.gif">。所以对应的正交特徵向量组<FONT FACE="Times New Roman">Q</FONT>为
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<IMG SRC="img00023.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00023.gif"><BR>
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在上述例子中，矩阵<FONT FACE="Times New Roman">A</FONT>很简单大小为<IMG SRC="img00024.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00024.gif">，可以用手做演算。一但矩阵大小增加，以<FONT FACE="Times New Roman">MATLAB</FONT>内建函数做运
算，就很轻松。相关函数的语法为<FONT COLOR=#FF0000 FACE="Times New Roman">eig(A)</FONT>，得到一栏向量代表<FONT COLOR=#FF0000 FACE="Times New Roman">A</FONT>的特徵值；而<FONT COLOR=#FF0000 FACE="Times New Roman">[Q,d]=eig(A)</FONT>，其中<FONT COLOR=#FF0000 FACE="Times New Roman">Q</FONT>代表<FONT COLOR=#FF0000 FACE="Times New Roman">A</FONT>的特徵
向量，<FONT COLOR=#FF0000 FACE="Times New Roman">d</FONT>为一对角矩阵其元素代表<FONT COLOR=#FF0000 FACE="Times New Roman">A</FONT>的特徵值。
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在此示范上述例子
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> A = [0.5 0.25;
0.25 0.5];</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> [Q,d] = eig(A)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">Q =</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 0.7071 0.7071</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -0.7071 0.7071</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">d = % </FONT><FONT COLOR=#FF0000>注意在对角线上的值才是特徵值</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 0.2500
0</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 0 0.7500
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> Q*Q</FONT><FONT COLOR=#FF0000>'
</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">% Q*Q</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">=I</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans=</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 1 0</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 0 1</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>> A*Q(:,1);
0.25* Q(:,1) % </FONT><FONT COLOR=#FF0000>验证<IMG SRC="img00025.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00025.gif">，注意</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">X=Q(:,1)
</FONT><FONT COLOR=#FF0000>为第一个特徵向量</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans = % </FONT><FONT COLOR=#FF0000>为</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">A*X</FONT><FONT COLOR=#FF0000>的结果</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 0.1768</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -0.1768</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans = % </FONT><FONT COLOR=#FF0000>为<IMG SRC="img00026.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img5/img00026.gif">的结果</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> 0.1768</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman"> -0.1768<BR>
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