# Eigenvalues & Eigenvectors

Algebra Level 3

True or false:

Let $$A:V\rightarrow V$$ and $$B:V\rightarrow V$$ be linear operators on a vector space $$V$$ over a field $$F$$.

If $$\mathbf{v}\in V$$ is an eigenvector of $$A$$ corresponding to $$\lambda\in F$$, then $$\mathbf{v}$$ is an eigenvector of $$BA-A-\lambda B$$ corresponding to $$-\lambda\in F$$, for any $$B$$.

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