Reflection matrix - are these two definitions equivalent?

Question

Are these two definitions of a reflection matrix Q equivalent (if and only if)?

Definition 1: $Q^TQ = I$ and $det(Q) = -1$

Definition 2: $Q = I-2nn^T$ where $n$ is a unit normal vector to the plane that reflects vectors.

It’s easy to show that definition 2 implies 1 (I’ve already done this - so please do not prove this part). But proving that definition 1 implies 2 seems much more difficult.

Answer 1

The result is wrong for real vector spaces with dimension $n>2$

The orthogonal transformation

$$Q= \begin{pmatrix} \dfrac{\sqrt{2}}{2} & -\dfrac{\sqrt{2}}{2} & 0\\ \dfrac{\sqrt{2}}{2} & \dfrac{\sqrt{2}}{2} & 0\\ 0 & 0 &-1 \end{pmatrix}$$

is such that $\det Q= - 1$. However, $Q$ is the not the matrix of a reflection ($Q$ has no fixed vector).

Answer 2

Take an odd permutation on $n$ symbols for large $n$, for example three parallel transpositions for $n=6$ such as (12)(34)(56). The corresponding permutation matrix will be orthogonal and have determinant $-1$.

Its eigen space corresponding to $-1$ eigenvalue is 3-dimensional and hence not a reflection matrix.

Much easier to see that diagonal matrices with $\pm1$'s in the diagonal are orthogonal and among them $-1$ determinant is easy to find again providing counterexamples.

Third kind is: get many $2\times2$ reflection matrices, say 3 of them. Call them $A,B,C$. Now construct a $6\times6$ matrix in block-diagonal form using $A,B,C$ as diagonal blocks.