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How to sample from a multivariate normal given the $P^TLDL^TP$ decomposition of $\Sigma$, and $\mu$?

Here, $P$ is a permutation matrix, $L$ is lower-triangular, and $D$ is diagonal

Given the Cholesky decomposition $LL^T$ of $\Sigma$, and $\mu$, we can sample by doing:

$sample = \mu + L * z$

... where $z$ is a vector of univariate standard normals.

I imagine that there must be some way of tweaking this formula to work with the $P^TLDL^TP$ decomposition, but my maths isn't quite good enough to see how to do that?

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    $\begingroup$ Perhaps your notation is confusing you, so let's write the decomposition as $P'\Lambda D\Lambda'P = LL'$. Now isn't it obvious that $L = P'\Lambda \sqrt{D}$ where the square root is applied to the diagonal elements of $D$? $\endgroup$
    – whuber ♦
    Jan 29, 2013 at 4:00
  • $\begingroup$ I thought that too, until I tried it. Unfortunately, it is not true that $L = P^T\Lambda \sqrt(D)$, because $L$ is lower-triangular, and $P^T\Lambda \sqrt(D)$ is not, because it's been permutated. $\endgroup$
    – Hugh Perkins
    Jan 29, 2013 at 4:11
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    $\begingroup$ But it's not necessary for $L$ to be lower triangular for this method to work! All that is required is that $LL'=\Sigma$. $\endgroup$
    – whuber ♦
    Jan 29, 2013 at 4:13
  • $\begingroup$ So you knew the answer all along :-). $\endgroup$
    – whuber ♦
    Jan 29, 2013 at 4:18
  • $\begingroup$ Well, I guessed at two plausible answers, but I lacked a way of proving/disproving either of them. $\endgroup$
    – Hugh Perkins
    Jan 29, 2013 at 4:21

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With credit to whuber, I'm just typing up the exchange in the comments so that this question is marked as having an answer.

We can write $$ P^\prime \Lambda D \Lambda^\prime P=LL^\prime $$

and $ L=P^\prime\Lambda\sqrt{D} $ where we understand $\sqrt{D}$ to be the square root of the diagonal of $D$. The only requirement for this to work is that $LL^\prime=\Sigma,$ so it's not important in this instance that $L=P^\prime\Lambda\sqrt{D}$ may be permuted.

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