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What is the eigenvalue/eigenvector relationship between matrix A,B and AB?

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closed as too broad by Will Jagy, Yemon Choi, Suvrit, Andrey Rekalo, Carlo Beenakker Jul 25 '13 at 10:21

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

I assume this is over $\mathbb C$. If they don't commute, about all you can say is that the determinant (which is the product of the eigenvalues, counted by algebraic multiplicity) of $AB$ is the product of the determinants of $A$ and $B$.

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I do not know about eigenvectors, but for the eigenvalues this is a special case of Deligne-Simpson Problem. It was completely solved by Crawley-Boevey about 10 years ago using quivers. For details, see my answer here and references therein.

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See the following paper. The review describes only results for $A+B$, but this readily transforms to $AB$, as described in the paper.

  • MR1957068 (2004b:14093) Reviewed Klyachko, Alexander(TR-BILK) Vector bundles, linear representations, and spectral problems. (English summary) Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 599–613, Higher Ed. Press, Beijing, 2002.

Review: The text provides a review of the amazing development resulting from the ground-breaking discovery of the author [Selecta Math. (N.S.) 4 (1998), no. 3, 419–445; MR1654578 (2000b:14054)] of the fundamental connections between inequalities for eigenvalues of Hermitian operators $A$, $B$ and $A+B$, Littlewood-Richardson coefficients $c^\gamma_{\alpha\beta}$ for the decomposition of the tensor product of irreducible $GL_n$-representations $$V_\alpha⊗V_β=∑c^γ_{αβ}V_γ,$$ the decomposition of the product of two Schubert cycles in the cohomology of the Grassmannian and toric stable vector bundles on P2. The topic has further ramifications for the groups $SL_n$, $SO_n$, $Sp_{2n}$ and other symmetric spaces. The paper contains no proofs but has an extensive list of references.

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Peter: Klyachko indeed solved the additive problem, but only for symmetric/hermitian matrices. This translates to solution of multiplicative problem, but not for eigenvalues --- you have to use singular values instead. Belkale solved multiplicative problem for unitary matrices. Solution for any other real forms of $SL(n,C)$ is currently unknown. – Misha Jul 25 '13 at 13:08

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