用MATLAB求矩阵特征值
用matlab求矩阵的特征值和特征向量
我要计算的矩阵：
1 3 5
1/3 1 3
1/5 1/3 1
[v,d]=eig(A);
A为你的矩阵，V为特征向量矩阵，D为特征值矩阵，然后对D求最大值即可得最大特征根！
[V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D.
V是特征向量，D是特征值
实例：
矩阵：
1 2/3 7/3 7/3
3/2 1 3/2 3/2
3/7 2/3 1 3/2
3/7 2/3 2/3 1
>> format rat
>> A=[1 2/3 7/3 7/3
3/2 1 3/2 3/2
3/7 2/3 1 3/2
3/7 2/3 2/3 1]
A =
1 2/3 7/3 7/3
3/2 1 3/2 3/2
3/7 2/3 1 3/2
3/7 2/3 2/3 1
>> [V,D]=eig(A)
V =
1793/2855 504/3235 - 146/235i 504/3235 + 146/235i 1990/4773
670/1079 -3527/5220 -3527/5220 -509/959
4350/11989 1160/4499 + 287/3868i 1160/4499 - 287/3868i -350/647
838/2819 181/3874 + 1179/4852i 181/3874 - 1179/4852i 1238/2467
D =
810/197 0 0 0
0 -93/4229 + 455/674i 0 0
0 0 -93/4229 - 455/674i 0
0 0 0 -149/2201
***************************************************************************************** 如何归一化求权重呢？
>> a=[1 3 5;1/3 1 3; 1/5 1/3 1]
a =
1.0000 3.0000 5.0000
0.3333 1.0000 3.0000
0.2000 0.3333 1.0000
>> [V,D]=eig(a)
V =
0.9161 0.9161 0.9161
0.3715 -0.1857 + 0.3217i -0.1857 - 0.3217i
0.1506 -0.0753 - 0.1304i -0.0753 + 0.1304i
D =
3.0385 0 0
0 -0.0193 + 0.3415i 0
0 0 -0.0193 - 0.3415i
**************************************************************************
>> a=[1 2 4 8 6 6 8;1/2 1 2 6 4 4 8;1/4 1/2 1 4 2 4 6;1/8 1/6 1/4 1 2 2 4;1/6 1/4 1/2 1/2 1 1 4;1/6 1/4 1/4 1/2 1 1 2;1/8 1/8 1/6 1/4 1/4 1/2 1]
a =
1.0000
2.0000 4.0000 8.0000 6.0000 6.0000 8.0000
0.5000 1.0000 2.0000 6.0000 4.0000 4.0000 8.0000
0.2500 0.5000 1.0000 4.0000 2.0000 4.0000 6.0000
0.1250 0.1667 0.2500 1.0000 2.0000 2.0000 4.0000
0.1667 0.2500 0.5000 0.5000 1.0000 1.0000 4.0000
0.1667 0.2500 0.2500 0.5000 1.0000 1.0000 2.0000
0.1250 0.1250 0.1667 0.2500 0.2500 0.5000 1.0000
>> rats(a)
ans =
1 2 4 8 6 6 8
1/2 1 2 6 4 4 8
1/4 1/2 1 4 2 4 6
1/8 1/6 1/4 1 2 2 4
1/6 1/4 1/2 1/2 1 1 4
1/6 1/4 1/4 1/2 1 1 2
1/8 1/8 1/6 1/4 1/4 1/2 1
>> [V,D]=eig(a)
V =
0.7884 0.8327 0.8327 0.8083 0.8083 -0.5119 + 0.3865i -0.5119 - 0.3865i
0.4894 0.3216 + 0.2636i 0.3216 - 0.2636i 0.1760 + 0.0792i 0.1760 - 0.0792i 0.6783 0.6783
0.3038 0.0883 + 0.2728i 0.0883 - 0.2728i -0.4630 + 0.1038i -0.4630 - 0.1038i -0.2011 - 0.2400i -0.2011 + 0.2400i
0.1404 -0.1620 + 0.1018i -0.1620 - 0.1018i 0.0620 - 0.0510i 0.0620 + 0.0510i -0.0006 + 0.1021i -0.0006 - 0.1021i
0.1215 -0.0627 - 0.0658i -0.0627 + 0.0658i 0.0367 + 0.2360i 0.0367 - 0.2360i -0.0531 + 0.0357i -0.0531 - 0.0357i
0.0975 -0.0303 - 0.0476i -0.0303 + 0.0476i 0.0488 - 0.1148i 0.0488 + 0.1148i 0.0231 - 0.1221i 0.0231 + 0.1221i
0.0508 0.0030 - 0.0590i 0.0030 + 0.0590i -0.0561 - 0.0454i -0.0561 + 0.0454i 0.0102 + 0.0197i 0.0102 - 0.0197i
D =
7.3899 0 0 0 0 0 0
0 -0.0008 + 1.5369i 0 0 0 0 0
0 0 -0.0008 - 1.5369i 0 0 0 0
0 0 0 -0.1624 + 0.6552i 0 0 0
0 0 0 0 -0.1624 - 0.6552i 0 0
0 0 0 0 0 -0.0317 + 0.2040i 0
0 0 0 0 0 0 -0.0317 - 0.2040i