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I have $$S = QLQ^T$$

I know $Q$, $L$, $Q^T$.

How can I get the $R$ and $R^T$ for the Cholesky decomposition $S=R^TR$?

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  • $\begingroup$ Isn't it simply $R = \sqrt{L} Q^T$? $\endgroup$ – vibe Mar 17 '20 at 5:36
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    $\begingroup$ It seems that it is not R is not upper triangular. $\endgroup$ – Jing Bai Mar 17 '20 at 5:40
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    $\begingroup$ Is $S$ a general matrix or it has some properties (like positive-definiteness)? $\endgroup$ – Anton Menshov Mar 17 '20 at 8:25
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    $\begingroup$ This looks a lot like an XY question. What do you need to do with the Cholesky decomposition that you cannot already do directly with the eigendecomposition? $\endgroup$ – Federico Poloni Mar 17 '20 at 10:07
  • $\begingroup$ positive definite symmetric $\endgroup$ – Jing Bai Mar 17 '20 at 23:18
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Let me re-write the eigen-decomposition as: $S = UDU^{\top}$. Now, consider the QR-decomposition $\sqrt{D}U^{\top} = QR$ where $Q$ is unitary and $R$ is upper triangular. For uniqueness, one has to ensure that $R$ has positive diagonal entries. For negative diagonal values, we can negate the corresponding row of $R$. $\,S=R^{\top}R$ is then the Cholesky decomposition.

Here is a MATLAB snippet to reproduce:

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)

If $S$ is full rank and positive definite, Cholesky decomposition is unique.

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  • $\begingroup$ Is it unique if S has degenerate eigenvalues given the arbitrariness of choice of linear combinations when forming U? $\endgroup$ – Ian Bush Mar 17 '20 at 13:31

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