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

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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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  • $\begingroup$ 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. $\endgroup$ – Misha Jul 25 '13 at 13:08

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