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blue2script
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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
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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