Problem: Given Gaussian RV X with Cov Matrix $C_{xx}$.
Find Matrix A such that Y is an RV, Y=AX and Cov Matrix of Y $C_{YY}=I$.
Now it's clear that according to some rule, $C_{YY}=AC_{XX}A^T$.
BUT $C_{YY}=I$, so A is diagonalizing!!!! Hence $C_{XX}=I$!!!!
I'll show you. Please help me find when I'm wrong.
$C_{YY}=AC_{XX}A^T$
$A^{-1}=A^T$ (Since A is diagonalizing) And hence -
$A^TC_{YY}A=C_{XX}$.
But $C_{YY}=I$ and $A^{-1}=A^T$ hence
$A^{-1}IA=C_{XX}$ =>
$C_{XX}=I$
What is the mistake? It's a common matrices whitenning problem.
I found the problem. It's the second time I make this mistake, hence I update for everyone:
$A^{-1}=A^T$ always holds true only if A is diagonolizing Mat X Orthogonally!!! If A is just a simple form of diaginolizing, it's not always true.