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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?

user83455
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    Many authors would use "reflection" to mean an orthogonal linear transformation with determinant $-1$, and you don't have enough to show that. E.g. the identity mapping. But this may be merely a technical issue, depending on the context in which you set up the problem. – hardmath Aug 22 '21 at 20:47
  • Such mappings can be characterized as follows: There exists a direct sum decomposition of the whole space, $V = V_1 \oplus V_2$ such that the mapping is the symmetry with respect to $V_1$ parallel to $V_2$. Concretely, what this means is that the image of $x = x_1 + x_2$ (where $x_i \in V_i$) is $x_1 - x_2$. Now whether this can be called a "reflection" depends on conventions, but many authors would require that $V_1$ and $V_2$ be orthogonal and perhaps even that $\dim V_2 = 1$. – Anonymous Aug 22 '21 at 21:21
  • I believe that some do talk about non-orthogonal reflections (in much the same way that people discuss non-orthogonal projections), in which case, what you describe here fits the bill. If there is no agreed inner product on the space, then one may always be defined so that such a transformation becomes an orthogonal reflection. You might also wish to consider matrices with only ${1}$ and ${-1}$ (i.e. $I$ and $-I$) as reflections, as they reflect in the whole space/trivial space respectively. You can characterise all such maps by the equation $R^2 = I$. If $R^* = R$ then $R$ is orthogonal. – Theo Bendit Aug 22 '21 at 22:35

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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!

user83455
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