On Fremdervectors: Vectors Orthogonal to Their Images Under Linear Transformations

Geometrically, the eigenvectors of a square matrix $\mathbf{A}$ are not rotated by $\mathbf{A}$. Here we consider vectors that are rotated $\pi \mathrm{/}2$ by $\mathbf{A}$; that is, vectors orthogonal to their images. We call these vectors fremdervectors of $\mathbf{A}$ and discuss conditions for their existence. We also define fremdervalues, scalars $z$ such that $z\mathbf{I}-\mathbf{A}$ has a fremdervector, and discuss several known applications for fremdervectors.