用matlab实现householderQR分解

以下是使用MATLAB实现Householder QR分解的示例代码：

function [Q,R] = householderQR(A)
% Householder QR分解
% 输入矩阵A，输出Q和R
% Q是正交矩阵，R是上三角矩阵

[m,n] = size(A);
Q = eye(m);
R = A;

for k = 1:n
% 计算Householder变换矩阵
v = zeros(m,1);
v(k:m) = R(k:m,k);
if v(k) >= 0
v(k) = v(k) + norm(v);
else
v(k) = v(k) - norm(v);
end
v = v / norm(v);
H = eye(m) - 2*v*v';

% 对R应用Householder变换
R(k:m,k:n) = H * R(k:m,k:n);

% 对Q应用Householder变换
Q(k:m,:) = H * Q(k:m,:);
end

end

% 示例
A = rand(5,3);
[Q,R] = householderQR(A);
Q*R % 验证QR分解是否正确