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Jul 25, 2013 at 13:08 comment added Misha 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.
Jul 25, 2013 at 9:23 history edited Peter Michor CC BY-SA 3.0
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Jul 25, 2013 at 5:22 history answered Peter Michor CC BY-SA 3.0