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Show that if a $3\times 3$ matrix $A$ represents the reflection about a plane, then $A$ is similar to the matrix $\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1 \end{bmatrix}$

This is not homework, just me preparing for a test -- but how can this be solved? I know the definition of similarity, but can't find a way to approach the problem.

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    @Jason: Please don't use the subject line as part of the message. The body should be self-contained, and the subject should describe the kind of thing you are asking about.2011-02-11

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Suppose that $A$ is a reflection about a plane spanned by $u,v$ and let $w$ be orthogonal to this plane, then $Au=u$ and $Av=v$ since the reflection through the plane doesn't change them and $Aw = -w$ (this is the reflection).

Now switch the basis to $u,v,w$ and you will get the result.


If you have any two diagonal matrices, that have the same entries on their diagonal, but maybe in a different order, then they are similar.

You can see this by defining the matrix $E_\sigma$ for a permutation $\sigma \in S_n$ on n elements as $(E_\sigma)_{i,\sigma(i)}=1$ and all the other entries are zero. If $D$ is diagonal then $E_\sigma D E_\sigma ^{-1}$ is just the permutation of the entries on the diagonal.