Question

This is mostly a reference request, as this must be well known!

Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$, which has the same eigenvalues) is similar to a symmetric matrix, so has real eigenvalues. Now take the vectors of eigenvalues of $A$ and of $B$, sorted in decreasing order, and let their componentwise product be $ab$. Then the question is: what is known about the relationship (for instance, inequalities) between $ab$ and the vector of eigenvalues of the product $AB$ (also taken in decreasing order). Some experimentation gives the conjecture that there is a majorization order between them, for instance. This must be well-known!

Answer 1

Here are the results that you are probably looking for.

The first one is for positive definite matrices only.

Theorem (Prob.III.6.14; Matrix Analysis, Bhatia 1997). Let $A$ and $B$ be Hermitian positive definite matrices. Let $\lambda^\downarrow(X)$ denote the vector of eigenvalues of $X$ in decreasing order; define $\lambda^\uparrow(X)$ likewise. Then, \begin{equation*} \lambda^\downarrow(A) \cdot \lambda^\uparrow(B) \prec \lambda(AB) \prec \lambda^\downarrow(A) \cdot \lambda^\downarrow(B) \end{equation*}

However, when dealing with matrix products, it is more natural to consider singular values rather than eigenvalues.

Therefore, the relation that you might be looking for is the log-majorization \begin{equation*} \log \sigma^\downarrow(A) + \log\sigma^\uparrow(B) \prec \log\sigma(AB) \prec \log\sigma^\downarrow(A) + \log\sigma^\downarrow(B), \end{equation*} where $A$ and $B$ are arbitrary matrices, and $\sigma(\cdot)$ denotes the singular value map.

Reference

1. R. Bhatia. Matrix Analysis. Springer, GTM 169. 1997.