# Simultaneous eigenspace of non-commuting matrices

• blue2script
In summary, the conversation discusses a brain teaser involving the simultaneous "eigenspace" of non-commuting matrices. The problem involves solving an equation for w_1 and w_2 using eigenvectors, and then extends to a more complex problem with three matrices where simultaneous eigenvectors cannot be found. A possible solution using an algorithm called SimDiag is suggested.

#### blue2script

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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## What is a simultaneous eigenspace?

A simultaneous eigenspace is a set of vectors that are eigenvectors of two or more non-commuting matrices. In other words, these vectors are able to simultaneously diagonalize all of the matrices in the set.

## Why is the concept of simultaneous eigenspace important?

The concept of simultaneous eigenspace is important because it allows us to simplify the diagonalization process for non-commuting matrices. By finding a set of eigenvectors that can simultaneously diagonalize all of the matrices, we can reduce the complexity of the problem and make it easier to solve.

## How do you find the simultaneous eigenspace of non-commuting matrices?

To find the simultaneous eigenspace of non-commuting matrices, you first need to find the common eigenvectors of the matrices. Then, you can use these common eigenvectors to construct a basis for the simultaneous eigenspace.

## Can the simultaneous eigenspace of non-commuting matrices be empty?

Yes, it is possible for the simultaneous eigenspace of non-commuting matrices to be empty. This occurs when the matrices do not have any common eigenvectors, meaning they cannot be simultaneously diagonalized.

## What is the relationship between simultaneous eigenspaces and commuting matrices?

If two matrices commute, then their simultaneous eigenspace is the same as their individual eigenspaces. However, if two matrices do not commute, their simultaneous eigenspace may not be the same as their individual eigenspaces. In fact, the simultaneous eigenspace may not even exist for non-commuting matrices.