I would like to know if there are any rare cases where performing a basis change for describing a transformation on a different basis also changes its eigenvalues.
I know that in general this does not happen, but I would like to know if there are any extreme cases where it does happen.
2026-04-01 23:11:59.1775085119
Are there any extreme instances where a change of basis can change the eigenvalues of the transformed matrix?
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No. If $T: V \rightarrow V$ is a linear operator where $A$ is the matrix of $T$ with respect to a basis $\mathcal{B}$, and $C$ is the matrix of $T$ with respect to another basis $\mathcal{B}^{'}$, then we necessarily have that $A=PCP^{-1}$ for some invertible matrix P. In other words, $A$ is conjugate (or similar) to $C$. Matrices that are similar always have the exact same eigenvalues.