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Q:

Do eigenvectors change when I change basis?

Let $T$ be a linear transformation on $\mathbb{R}^3$, and suppose that $T$ has a eigenvalue $\lambda$ with eigenvector $u$. If I consider the linear transformation $T\circ J$, where $J$ is the matrix you get when you simultaneously rotate the basis of $\mathbb{R}^3$ by an angle $2\pi/3$ clockwise, then $T\circ J$ has an eigenvalue $\lambda$ with eigenvector $Ju$.
Question: In terms of linear transformations, if $T$ and $T\circ J$ have eigenvalues $\lambda$ and $\lambda$, does it follow that $u$ is also an eigenvector of $T$?
I’m inclined to think this is true, because the determinant of $J$ is $-1$, and $u$ is an eigenvector of $T\circ J$ with the same eigenvalue as $T$, so the eigenvalues and eigenvectors of $T\circ J$ are related to the eigenvalues and eigenvectors of $T$.
Thanks for any help.

A:

Note that
$$
T J = J T \iff JT – TJ = 0
$$
Take
$$
T=\pmatrix{2&0&0\\0&1&0\\0&0&-1}
$$
and
$$
T J = \pmatrix{1&-1&0\\0&-1&-1\\0&0&1}.
$$
All eigenvalues are $1$ because $\det(T)=1$. The eigenvectors are
$$
v_1 = \pmatrix{1\\0\\1}, \ v_2

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