TITLE: Eigenvalues of product of matrices
QUESTION [7 upvotes]: If $\mathbf{A}_{n\times n}$ is a positive semi-definite matrix with eigenvalues $\{\alpha_k\},\ k\in\{1,...,n\}$, and $\mathbf{B}_{m\times n}$ is an arbitrary matrix with singular values $\{\beta_k\},\ k\in\{1,...,\min(m,n)\}$, can anything be said about the singular values $\{\gamma_k\},\ k\in\{1,...,\min(m,n)\}$ of the matrix $\mathbf{\Gamma}=\mathbf{BA}$? Is there a way I can relate $\gamma_k$ to $\alpha_k$ and $\beta_k$?

REPLY [9 votes]: The singular values of $B$ are the square roots of the eigenvalues of $B^* B = C$, and those of
$BA$ are the square roots of the eigenvalues of $(BA)^* BA = A C A$.  So the question is whether for positive semidefinite matrices $A$ and $C$ there is any relation between eigenvalues of $A C A$ and those of $A$ and $C$.  The answer is, in general, not much.
You might consider some simple $2 \times 2$ examples, with different $C$ having the same eigenvalues, but the corresponding $A C A$ having different eigenvalues.
Of course the product of the eigenvalues is the determinant, and $\det(ACA) = \det(C) \det(A)^2$.  Also, the sum of the eigenvalues is the trace, and using Cauchy-Schwarz you have a bound
${\rm tr}(ACA) = {\rm tr}(CA^2) \le ({\rm tr}( C^2))^{1/2} ({\rm tr}(A^4))^{1/2}$.