Find the signature of a reflection.

Question

I am doing some self-study and I have this task:

Show that the signature of the following mapping, the reflection about the hyperplane $a^{\perp}$, given by

$S_a(v) = v - 2\frac{<v,a>}{<a,a>}a$

is (n-1,1,0), where a is an arbitrary nonzero vector.

I would know how to find the eigenvalues of this mapping if I had a matrix on hand, but I am not sure how I would do it this way. Should I just find the representative matrix of this mapping and go with it?

Answer 1

Under the reflection $a\to -a$, $a^{\perp}$ is invariant. In any basis with this split $e_1=a, e_{2\dots n} \in a^{\perp}$, the signature is evident.

Answer 2

A reflection is a special case of linear involution, with a hyperplane as subspace of fixed vectors: $S_a(v)=v\iff v\perp a$ and $S_a(v)=-v\iff v\in\operatorname{span}(a)$.

More generally (and even on an infinite dimensional space), a linear involution has two complementary eigenspaces, associated with the eigenvalues $1$ and $-1$.