John
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 Jan 30 comment Efficient Cholesky decomposition of inverse matrix I'm doing the same thing, but I have been getting funky results. You can take the cholesky decomposition of $\Sigma^{-1}$ and find the upper cholesky decomposition of that and then take the inverse. However, what I am left with is a triangular matrix that can reproduce $\Sigma$, but it isn't the same as $chol(\Sigma)$ Jan 28 comment How to generate a random matrix whose eigenvalues are less than one I've been beating my head against the wall trying to do a similar thing. My application is ensuring Bayesian VAR results are stationary in a MCMC, which I think is a slightly different set up. Sep 12 comment Finding $X$ When $Y=XX'$ @BaronVT I see your point now. What kinds of conditions would need to be made on the eigenvalues. In the particular ones I'm working with, it will often be the case that several will be equal to each other. Sep 12 comment Finding $X$ When $Y=XX'$ @MichaelHardy Thanks. Sep 11 comment Finding $X$ When $Y=XX'$ $Y$ is some known matrix. It could be an identity matrix, or it could be anything else so long as it is positive definite.