# Simultaneous eigenspace of non-commuting matrices

## Main Question or Discussion Point

Simultaneous "eigenspace" of non-commuting matrices

Hello!

I have been working on the following "brain teaser" the whole day long without any success. I am not even sure there is a "clean" solution. I would love to hear your opinion. Before presenting the whole problem, here is an easy pre-stage: Solve

$$w_{2}^{T}\left(C_{1}-C_{2}\right)w_{1} = 0$$

for $$w_1, w_2\in R^n, w_1^Tw_2 = 0, w_i^Tw_i = 1$$ and given $$C_1, C_2\in R^{n\times n}$$. Here $$C_1, C_2$$ are covariance matrices (symmetric and positive semidefinite). This equation is solved by the eigenvectors of $$D = C_1 - C_2$$.

Now, the full problem is this:

$$w_{2}^{T}\left(C_{2}-C_{1}\right)w_{1} = 0$$
$$w_{3}^{T}\left(C_{3}-C_{1}\right)w_{1} = 0$$
$$w_{3}^{T}\left(C_{3}-C_{2}\right)w_{2} = 0$$

where $$w_{i}, C_i$$ have the same properties as before (and all $$w_i$$ are orthogonal and of unit length). However, $$C_i, C_j$$ nor their sums commute and we can't find simultaneous eigenvectors of all the three matrices. I tried to solve this problem by using simultaneous singular value decomposition (see the paper from Takanori Maehara and Kazuo Murota), but I couldn't get it to work.

Any hints are highly appreciated. Thanks a lot!
blue2script

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Hi,

there is an algorithm for the computation of (almost) eigenvectors for a set of matrices implemented in ApCoCoA http://www.apcocoa.org/. It's called SimDiag.
http://www.apcocoa.org/wiki?title=ApCoCoA:Num.SimDiag [Broken]

Hope this helps!

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