matlab QR分解 求矩阵的特征值

Posted 2020-11-09 金舰

"QR_H.m"

function [Q,R] = QR_tao(A)
%输入矩阵A
%输出正交矩阵Q和上三角矩阵R
[n,n]=size(A);
E = eye(n);
X = zeros(n,1);
R = zeros(n);

P1 = E;
for k=1:n-1
    s = -sign(A(k,k))*norm(A(k:n,k));
    R(k,k) = -s;
    if k == 1
        w = [A(1,1)+s,A(2:n,k)\']\';
    else
        w = [zeros(1,k-1),A(k,k)+s,A(k+1:n,k)\']\';
        R(1:k-1,k) = A(1:k-1,k);
    end
    if norm(w)~=0
        w = w/norm(w);
    end
    P = E-2*w*w\';
    A = P*A;
    P1 = P*P1;
    R(1:n,n) = A(1:n,n);
end

 

之后根据算法:

An = Q1*R1;

An+1 =  R1*Q1

重复迭代即可。

"QR.m"

%输入 矩阵A 和迭代次数 it_max
%输出 最后对角线上元素为特征值的矩阵
function [Q] = QR(A,it_max)
A1 = A;
for N=1:it_max
[Q1,R1] = QR_tao(A1);
A2 = R1*Q1;
A1 = A2;
end
Q=A1

测试： 计算一个矩阵的特征值:

A = [13,-3,-1,0;
    -3,6,0,-2;
    -1,0,10,-1;
    0,-2,-1,7;
];
[Q] = QR(A,50)
eig(A)

 

最后结果: