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For a particular engineering problem that I'm working on, I have computed a Jacobian matrix $J$ and there is another matrix $M$ associated with the problem. $M$ is known to be symmetric, real-valued, and positive definite. At a particular step in the process, I need to compute the eigenvalues of $MJ$.

I am wondering if there are any known results about how the eigenvalues of $MJ$ relate (via shifting, scaling, or other transformations) to the eigenvalues of $J$, if at all. Statements that require $M$ to have certain extra properties would be valuable too (i.e. I realize there are dumb corner cases such as when $M=J^{-1}$, for example, so answers that meaningfully exclude cases like that but which leave open interesting results are welcome, if they exist).

Note that I don't mean the classical problem of simultaneous diagonalization. This is really a computational problem at root. I'm trying to avoid needing to do a more complicated numerical solution for the eigenvalues of $MJ$ if possible. In my program, I will already have pre-computed the eigenvalues of $J$ and it would lose efficiency if, after getting the associated matrix $M$, I had to then solve the classical problem of computing the spectrum of $MJ$. The goal is make the overall numerical method faster by exploiting any knowledge that $J$ gives us about the spectrum of $MJ$.

In trying to think about this, we can assume that $J$ yields an eigenbasis of $\{\lambda_{k},e_{k}\}$, so that $MJx = \sigma{x}$ can be rewritten $\sum_{k}\lambda_{k}Me_{k} = \sum_{k}\sigma\lambda_{k}e_{k}$ for any $x$ that happens to be an eigenvector of $MJ$.

What kinds of situations then allow us to make statements about $\sigma$ in terms of the $\lambda_{k}$, especially for somewhat large classes of matrices $M$?

The references that I have already looked through are "Matrix Analysis" by Horn and Johnson, Gil Strang's Linear Algebra book, and "Matrix Computations" by Golub and Van Loan, none of which gives any kind of usable answer.

References to research papers or books that shed any light would be appreciated if (as I suspect) this turns out to be a question that's not really answerable in general and statements can only be made for narrow classes of matrices $M$.

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