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## 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

[tex]w_{2}^{T}\left(C_{1}-C_{2}\right)w_{1} = 0[/tex]

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

Now, the full problem is this:

[tex]w_{2}^{T}\left(C_{2}-C_{1}\right)w_{1} = 0[/tex]

[tex]w_{3}^{T}\left(C_{3}-C_{1}\right)w_{1} = 0[/tex]

[tex]w_{3}^{T}\left(C_{3}-C_{2}\right)w_{2} = 0[/tex]

where [tex]w_{i}, C_i[/tex] have the same properties as before (and all [tex]w_i[/tex] are orthogonal and of unit length). However, [tex]C_i, C_j[/tex] 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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