Suppose I have a diagonalizable real matrix $A$ with the right eigenvectors $X$ and diagonal eigenvalue matrix $\Lambda$ such that $$A = X \Lambda X^{-1}.$$ (In fact in my case $A$ is symmetric so $X^{-1} = X^T$). Suppose I also have a known matrix $B$ (in my base $B$ is real skew-symmetric), and I am interested in looking at properties of the matrix $$A_2 = X B \Lambda X^{-1}.$$ Is there anything I can say about $A_2$ related to $A$? I know it's not a well formed question, but since I'm not sure if there is any established theory on this, I tried googling but lacked the proper terminology to find anything meaningful.
2026-04-08 05:34:17.1775626457
Linear transformation on eigenvalues within an eigenvalue decomposition
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