On my other question users have been kindly helping and one has said "any diagonalizable matrix with eigenvalues ±1 is a reflection". I want to confirm if this is correct terminology, it seems unintuitive to me, but I am not an expert.
For example is $\begin{pmatrix}-1 & 2 \\ 0 & 1 \end{pmatrix}$ considered a reflection? Perhaps there is confusion also about which inner product is being used. Is it correct to say reflection in some basis, but not necessarily in the active basis?
As with many terminology questions, it seems the answer has some variability depending on context.
If we take the definition of "reflection" to come from everyday real world mirrors, then it is a matrix with an orthogonal basis of eigenvectors having eigenvalues {1, 1, -1} (exactly one -1 eigenvalue in any number of dimensions).
This can generalised this to include "reflections" through a point, through a line etc. in which case any of the eigenvalues can be negative. It can also be generalised to "non-orthogonal reflections" where the direction of reflection is not perpendicular to the mirror, in which case the eigenvectors do not have to be orthogonal.
"any diagonalizable matrix with eigenvalues ±1 is a reflection" if "reflections" include both of these generalisations.
This answer is based fully from the commenters, thanks to you!