简单易懂！使用幂法在Matlab中找到矩阵的主特征值与对应特征向量教程

题目

用幂法计算下列矩阵的按模最大特征值及对应的特征向量

幂法

代码

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 简介：用幂法计算矩阵的主特征值和对应的特征向量
% 作者：不雨_亦潇潇
% 文件：mifa.m
% 日期：20221109
% 博客：https://blog.****.net/weixin_43470383
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clc; clear all;
% A = [7 3 -2; 3 4 -1; -2 -1 3];
A = [3 -4 3; -4 6 3; 3 3 1];
u =[1; 1; 1];
err = 10^(-4);
% 第一次迭代
k = 1;
m0 = 1;
V =A*u;
m = max(V);
u = V/m;
disp('k=')
disp(k)
disp('V^T=')
disp(V')
disp('m=')
disp(m)
disp('u^T=')
disp(u')
while abs(m0-m)>err
    m0 = m;
    k = k+1;
    V =A*u;
    m = max(V); % V绝对值最大的分量
    u = V/m; % 规范化向量  
    disp('k=')
    disp(k)
    disp('V^T=')
    disp(V')
    disp('m=')
    disp(m)
    disp('u^T=')
    disp(u')
end
lambda = m0 % 最终得到的主特征值
vector = u % 最终得到的特征向量

题目1输出结果

k=

1

V^T=

8 6 0

m=

8

u^T=

1.0000 0.7500 0

k=

2

V^T=

9.2500 6.0000 -2.7500

m=

9.2500

u^T=

1.0000 0.6486 -0.2973

k=

3

V^T=

9.5405 5.8919 -3.5405

m=

9.5405

u^T=

1.0000 0.6176 -0.3711

k=

4

V^T=

9.5949 5.8414 -3.7309

m=

9.5949

u^T=

1.0000 0.6088 -0.3888

k=

5

V^T=

9.6041 5.8240 -3.7753

m=

9.6041

u^T=

1.0000 0.6064 -0.3931

k=

6

V^T=

9.6054 5.8187 -3.7857

m=

9.6054

u^T=

1.0000 0.6058 -0.3941

k=

7

V^T=

9.6056 5.8172 -3.7881

m=

9.6056

u^T=

1.0000 0.6056 -0.3944

k=

8

V^T=

9.6056 5.8168 -3.7887

m=

9.6056

u^T=

1.0000 0.6056 -0.3944

lambda =

9.6056

vector =

1.0000

0.6056

-0.3944

题目2输出结果

k=

1

V^T=

2 5 7

m=

7

u^T=

0.2857 0.7143 1.0000

k=

2

V^T=

1.0000 6.1429 4.0000

m=

6.1429

u^T=

0.1628 1.0000 0.6512

k=

3

V^T=

-1.5581 7.3023 4.1395

m=

7.3023

u^T=

-0.2134 1.0000 0.5669

k=

4

V^T=

-2.9395 8.5541 2.9268

m=

8.5541

u^T=

-0.3436 1.0000 0.3421

k=

5

V^T=

-4.0045 8.4010 2.3112

m=

8.4010

u^T=

-0.4767 1.0000 0.2751

k=

6

V^T=

-4.6047 8.7320 1.8451

m=

8.7320

u^T=

-0.5273 1.0000 0.2113

k=

7

V^T=

-4.9481 8.7432 1.6293

m=

8.7432

u^T=

-0.5659 1.0000 0.1864

k=

8

V^T=

-5.1387 8.8228 1.4886

m=

8.8228

u^T=

-0.5824 1.0000 0.1687

k=

9

V^T=

-5.2412 8.8359 1.4214

m=

8.8359

u^T=

-0.5932 1.0000 0.1609

k=

10

V^T=

-5.2969 8.8553 1.3814

m=

8.8553

u^T=

-0.5982 1.0000 0.1560

k=

11

V^T=

-5.3265 8.8606 1.3615

m=

8.8606

u^T=

-0.6011 1.0000 0.1537

k=

12

V^T=

-5.3425 8.8655 1.3502

m=

8.8655

u^T=

-0.6026 1.0000 0.1523

k=

13

V^T=

-5.3509 8.8673 1.3445

m=

8.8673

u^T=

-0.6034 1.0000 0.1516

k=

14

V^T=

-5.3555 8.8686 1.3413

m=

8.8686

u^T=

-0.6039 1.0000 0.1512

k=

15

V^T=

-5.3579 8.8692 1.3396

m=

8.8692

u^T=

-0.6041 1.0000 0.1510

k=

16

V^T=

-5.3592 8.8695 1.3387

m=

8.8695

u^T=

-0.6042 1.0000 0.1509

k=

17

V^T=

-5.3599 8.8697 1.3383

m=

8.8697

u^T=

-0.6043 1.0000 0.1509

k=

18

V^T=

-5.3602 8.8698 1.3380

m=

8.8698

u^T=

-0.6043 1.0000 0.1509

lambda =

8.8697

vector =

-0.6043

1.0000

0.1509