**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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