Is there a way to solve the matrix equation $XX^* = A$, where $X$ is a $n\times k$ unknown matrix and $A$ is a $n\times n$ positive-definite Hermite matrix?
Cholesky decomposition may be useful when $n=k$, but how about the case where $n \neq k$?
Thanks so much!
Start with the Cholesky decomposition. That is, for any such $A$, we have an $L$ such that $LL^* = A$. Now, in the case that $k>n$, construct the $n\times k$ block matrix $X$ as follows: $$ X = \begin{bmatrix} L & 0 \end{bmatrix} $$ Where $0$ here is the $n\times(k-n)$ zero matrix. We find that $$ XX^* = \begin{bmatrix} L & 0 \end{bmatrix} \begin{bmatrix} L^* \\ 0^* \end{bmatrix}= LL^* + 00^* = LL^* = A $$ This will not generally be possible in the case that $k<n$. For a given symmetric, positive semidefinite matrix $A$, if $XX^* = A$ and $X$ is $n\times k$, then $k$ must be greater than or equal to the rank of $A$.