In this question, I understand how to get the eigenvalues, but I don't really understand what they mean (in this context of a reflection). I'm mostly looking for how an eigenvalue of -1 means we have to take the transpose of the matrix here. Thanks!
2026-03-27 02:39:52.1774579192
Geometric interpretation of eigenvalues for a reflection
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$\newcommand{\Vec}[1]{\mathbf{#1}}$In the hope a diagram helps, here's a generic decomposition $\Vec{v} = \Vec{v}_{1} + \Vec{v}_{-1}$ of a vector into eigenvectors of a reflection, and their respective images.