Given a random gaussian matrix X with zero mean matrix and covariance matrix Σ, and two deterministic matrices A and B. If I know the value of $||{\bf{AX}}||_F^2$, how could I get the pdf of $||{\bf{BX}}||_F^2$ ($||{\bf{P}}||_F^2$ is the L2 norm of matrix P)?
In my opinion, since X is a random gaussian matrix, $||{\bf{AX}}||_F^2$ follows chi-square distribution and its pdf can be calculated. However, if I obtain the value of the random variable $||{\bf{AX}}||_F^2$ without knowing specific X, the pdf of X and the pdf of $||{\bf{BX}}||_F^2$ are not obvious.
What I want to know is, what are the distributions of X and $||{\bf{BX}}||_F^2$ given $||{\bf{AX}}||_F^2=a$? Or, is there any method to get the pdf of $||{\bf{BX}}||_F^2$ given $||{\bf{AX}}||_F^2=a$ without calculating the pdf of X.
I have learned the pdf of X given S=s, where S is the sum of all elements of X Distribution of joint Gaussian conditional on their sum. But it seems quite different from this question.