I'm studying multivariate distributions in general and I keep coming across an expression like:
$X = \mu + A Y$ where $ \mu $ and $A $ are constant vectors of dimensions d x 1 and d x k respectively. Then $X$ is d x 1 and $Y$ k x 1.
Then theres an expression saying $ \Sigma = AA'$ and my lecturer mentions how we get $A$ via cholesky decomposition. Here's what I dont understand. if $A$ is d x k , i.e. not necessarily a square matrix, how can it be derived using cholesky if cholesky only produces square matrices, at least I think it does. It also seems strage to me that k can be larger or smaller then d, but I think it makes some sense. For example say k was 1, in which case $X$ would be generated from a single univarite distrbution $Y$. Or if d was 2 and k was 5 then $X$ would be a bivarite distribution generated from a linear combination of those distributions making up $Y$ of dimension 5... This second part, which is a bit fuzzy in my head, I think is okay but I really am struggling to understanding how $A$ can be non-square if it's derived using choleskly decomposition.