A is a diagonal matrix and B is a symmetric matrix, both square matrices of same order. All elements in A and B are positive. I am trying to find a formulation for the largest eigenvalue of their product, AB.

I checked related answered questions (here, here, and here) but they seem to pertain to positive definite or positive semi definite matrices. In my case, A and B need not be positive semi definite.

The best I can do till now is getting a bound on the largest eigenvalue of AB using the Perron-Frobenius theorem, since AB is positive. However, I am searching for a way to do better than an upper bound, if it's possible, and get an exact expression for the dominant eigenvalue.

  • $\begingroup$ A diagonal matrix with positive entries on the diagonal is positive definite. Your second link applies. $\endgroup$ – bangs Sep 3 '18 at 0:24
  • $\begingroup$ @bangs Thank you, I didn't recognize A as positive definite. However, the second link still just talks about the number of eigenvalues. It does not say anything about the largest eigenvalue, and how I can find a formulation for that. $\endgroup$ – Learner Yoda Sep 3 '18 at 0:36
  • $\begingroup$ Sorry, I talked about the third link in my comment above. However, even in the second link, the answer just talks about a lemma which can provide a lower bound of the largest eigenvalue. $\endgroup$ – Learner Yoda Sep 3 '18 at 0:48
  • $\begingroup$ Let $C$ be the obvious positive diagonal square root of $A$. Then the eigenvalues of $AB$ are the same as the eigenvalues of $CBC$, which is also a symmetric positive matrix. So if you know how to answer the seemingly simpler question involving 1 of your special matrices you know how to answer the question about 2. The power method can often give useful upper & lower bounds of the max eigenvalue. $\endgroup$ – kimchi lover Sep 3 '18 at 1:19
  • $\begingroup$ Thanks for your comment. I understand now that instead of having to find the eigenvalues for AB, I can now just find the eigenvalues of CBC = D (let's say). However, I only know of how to get a lower bound for the largest eigenvalue for a real symmetric positive matrix, as $\lambda_{max} \geq v'Dv$, where $||v|| = 1$. Is there a way to find the exact formulation for the largest eigenvalue for a symmetric positive matrix? Also, what do you mean by the power method? $\endgroup$ – Learner Yoda Sep 4 '18 at 2:53

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