Suppose we have a normally distributed random variable $\boldsymbol{X}$ with covariance matrix $\Sigma$, which is symmetric positive definite.
Now if I multiply $\boldsymbol{X}$ by some matrix A, the resulting random variable has covariance:
cov($A \boldsymbol{X}$) = $A\Sigma A'$.
Is there any restrictions on the matrix A such that $A\Sigma A'$ is a valid covariance, ie is symmetric positive definite with full rank and non zero determinant?
Edit:
$\Sigma$ is a diagonally dominant matrix of size 10x10, with 6 in the diagonal elements and -1's in 5 of the columns for each row.
Matrix A contains data and is large 480x10. I have tried using just random generated normal data in A and get the same issue that $A\Sigma A'$ has rank 10 and zero determinant. The rank of the resulting matrix seems to depend on the rank of $\Sigma$.