Let $S$ and $T$ be two linear transformations on $\mathbb{R}^n$. We can find the eigenvalues and eigenvectors of $S$ and $T$. I am trying to find out the relation(s) among the eigenvalues and eigenvectors of $S$, $T$ and $S\circ T$. Is there any relation of this kind?
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3$\begingroup$ In gebera this seems hopeless. If they are normal and commutte, then easy. Do you know anything about your matrices? $\endgroup$– András BátkaiCommented Aug 21, 2019 at 12:28
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$\begingroup$ @AndrásBátkai excelent comment! An infinite dimensional analogy is that a commuting familly of compact operators has a common invariant subspace. $\endgroup$– Ali TaghaviCommented Aug 21, 2019 at 12:39
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$\begingroup$ @AndrásBátkai what do you mean by 'gebera', sorry? $\endgroup$– vidyarthiCommented Aug 21, 2019 at 12:56
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5$\begingroup$ I am guessing he meant 'general'. $\endgroup$– R.P.Commented Aug 21, 2019 at 13:18
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3$\begingroup$ For products of unitary matrices, there are inequalities relating the eigenvalues of the product with that of the factors, see arxiv.org/abs/alg-geom/9712013 . For arbitrary matrices, there is not much one can say beyond what one can get from the determinant identity $\det(AB) = \det(A) \det(B)$. $\endgroup$– Terry TaoCommented Aug 21, 2019 at 16:34
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1 Answer
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This is a very difficult problem, the relations are not equalities but inequalities, and in an important special case it was solvedd in
S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, math.AG/9712013, Math. Res. Lett. 5 (1998), 817–836. MR 2000a:14066
answered Aug 21, 2019 at 20:52