Eigenvalues of AX - XA: Finding Eigenvectors for a Real 2x2 Symmetric Matrix

[alchemik]

Hi all,

Here is this problem that I have been at for some time now: find eigenvalues and corresponding eigenvectors of the following linear mapping on a vector space of real 2 by 2 matrices:

$$L(X) = AX - XA$$, where A is 2 by 2 symmetric matrix that is not a scalar multiple of identity.

It is clear that 0 is one eigenvalue of the above with an eigenspace consisting of all matrices X that commute with A, I do not know however what its algebraic multiplicity would be, except that it has to be less than 4. To find the other eigenvalues I looked at

$$AX - XA = \lambda X$$

a column at a time to obtain a 4 by 4 system.

$$Ax_{i} - \sum^{2}_{j = 1}a_{ji}x_{i} = \lambda x_{i}$$

for i = 1, 2, with j indexing the rows of A.

This approach requires me to find eigenvalues and eigenvectors of a 4 by 4 matrix, which may get messy if you have to do it by hand. Do you guys know any other, more efficient way of approaching this problem? I have attempted to use the fact that A can be orthogonally diagonalized in hopes of simplifying the above but with no success.

Thanks.

 

[adriank]

Diagonalize A: let A = P^-1DP, where D = diag(k₁, k₂). Given any matrix X, write
$$X = P^{-1} \begin{pmatrix}a&b\\c&d\end{pmatrix}P.$$
You can compute that
$$L(X) = P^{-1} \begin{pmatrix}0&(k_1-k_2)b\\(k_2-k_1)c&0\end{pmatrix}P.$$
You can now directly read off the eigenvalues of L: 0, 0, k₁ - k₂, and k₂ - k₁. The eigenvectors are also obvious.

The same process applies to arbitrary diagonalizable matrices. If A is a n × n diagonalizable matrix and L(X) = AX - XA, and A has eigenvalues k₁, ..., k_n, then the eigenvalues of L are k_i - k_j, for each pair of i, j from 1, ..., n.

 

[alchemik]

This is indeed a lot more tractable, thanks a lot adriank!