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 $M$.


1 Answer 1


I know this was asked a long time ago, but I think I have a useful way of thinking about this problem. Observe that $M=LL^{T}$ where $L$ is a nonsingular lower triangular matrix (the Cholesky factorization). Then $MJ=LL^{T}J,$ and $L^{-1}MJL=L^{T}JL.$ That is, $MJ$ is similar to $L^{T}JL,$ so they have exactly the same eigenvalues. Now observe that $L^{T}JL$ is congruent to $J,$ so by a theorem of Ostrowski (which can be found in Horn and Johnson), if the eigenvalues are put into increasing order for both $J$ and $L^{T}JL$, we have $$\lambda_{k}(L^{T}JL)=\theta_{k}\lambda_{k}(J),$$ where $\theta_{k}\in[\lambda_{1}(LL^{T}),\lambda_{n}(LL^{T})]=[\lambda_{1}(M),\lambda_{n}(M)]$ for each $1\leq k\leq n.$ Then if we can get some information about the eigenvalues of $M,$ in particular the first and last eigenvalues of $M$ or some estimates for these, call them $\tilde{\lambda}_{1}$ and $\tilde{\lambda}_{n},$ we have a good starting point for the shifted-inverse power method for computing $\lambda_{k}(L^{T}JL)$ ($L^{T}JL$ is easier to work with than $MJ$ since it is Hermitian) given by $\left(\frac{\tilde{\lambda}_{1}+\tilde{\lambda}_{n}}{2}\right)\lambda_{k}(J)$.

Note that these estimates do not actually require us to compute $M=LL^{T},$ since $\tilde{\lambda}_{1},\tilde{\lambda}_{n}$ depend only on $M,$ and $\lambda_{k}(MJ)=\lambda_{k}(L^{T}JL)$ because these matrices are similar. Also, depending on how close $\lambda_{1}(M)$ and $\lambda_{n}(M)$ are and how good you require your estimates for the eigenvalues of $MJ$ to be, it might be possible to just use $\left(\frac{\lambda_{1}(M)+\lambda_{n}(M)}{2}\right)\lambda_{k}(J)$ as your estimates for $\lambda_{k}(MJ)$ (you'd probably want to compute near-exact values for $\lambda_{1}(M)$ and $\lambda_{n}(M)$ in this case, which you could do with power method for the top eigenvalue and power method on $M^{-1}$ for the smallest eigenvalue (and using an initial eigenvector guess orthogonal to the approximated top eigenvector for faster convergence)).

Since we have estimates for all of the eigenvalues of $MJ$ (or $L^{T}JL$), we could use shifted-inverse power method with successive deflations for computing all of the eigenvalues of $MJ,$ but perhaps there is some way to "warm-start" the QR algorithm to get faster convergence using these estimates. Some discussion of such an idea is given here, but some additional searching could prove fruitful.


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