让我将特征分解重写为：。现在，考虑 QR 分解，其中是单一的，是上三角形。为了唯一性，必须确保具有正对角线条目。的相应行。是 Cholesky 分解。$S = UDU^{\top}$$\sqrt{D}U^{\top} = QR$$Q$$R$$R$$R$$\,S=R^{\top}R$

这是一个要重现的 MATLAB 代码段：

n = 5;
A = rand(n); 

% construct S to be positive-definite:
S = A'*A; 
S = S + n*eye(n);
[U,D] = eig(S); % we assume this to be given.

% original cholesky for comparisons
Rchol = chol(S);

% perform the proposed Cholesky
[~,Rqr] = qr(sqrt(D) * U');
Dg = diag(sign(diag(Rqr)));
Rqr = Dg * Rqr;

% now verify that Rqr = Rchol are the same
disp(['error in R: ' num2str(norm(Rchol-Rqr))]);

% now verify that S can be reconstructed from Rqr
disp(['reconstruction error: ' num2str(norm(S - Rqr'*Rqr))]);

% both results should be almost 0 (up to the numerical precision)

如果是满秩且正定的，则​​ Cholesky 分解是唯一的。$S$