Why to maximize the determinant of $A \Sigma_x A^T$ requires the columns of A be the eigenvectors of X?

Question

Let x be Gaussian with covariance $\Sigma_x$, and A an matrix with all the columns being unit vectors.

Then how can I prove that if I want to maximize det($A\Sigma_xA^T$), I can just set the columns of A to be the eigenvectors of X?

Answer 1

$\det(A\Sigma A^T)=\det(A)^2\det(\Sigma)$, so you just want to maximize $\det(A)$. Can you see how to conclude?

Try proving the following lemma.

Lemma. Let $A$ be a matrix whose columns have norm $\leq1$. Then $\det(A)\leq1$, with equality if and only if $A$ is orthogonal.

Hint: Gram-Schmidt...