On Fremdervectors: Vectors Orthogonal to Their Images Under Linear Transformations

TL;DR

This paper introduces the concept of fremdervectors, vectors orthogonal to their images under a linear transformation, exploring their properties, existence conditions, and applications in linear algebra.

Contribution

It defines fremdervectors and fremdervalues, providing conditions for their existence and discussing their relevance in various applications.

Findings

Fremdervectors are vectors rotated by π/2 under a linear transformation.

Conditions for the existence of fremdervectors are established.

Applications of fremdervectors are discussed in linear algebra contexts.

Abstract

Geometrically, the eigenvectors of a square matrix $\mathbf{A}$ are not rotated by $\mathbf{A}$. Here we consider vectors that are rotated $\pi/2$ 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.