Beside the
eigenspace
for
,
which is the
kernel
of the linear mapping, the eigenvalues
and
are in particular interesting. The eigenspace for consists of all vectors which are sent to themselves. Restricted to this linear subspace, the mapping is just the identity, it is called the fixed space. The eigenspace for consists in all vector which are sent to their negative. On this linear subspace, the mapping acts like the reflection at the origin.
We prove the statement by induction on . For
,
the statement is true. Suppose now that the statement is true for less than vectors. We consider a representation of , say
We apply to this and get, on one hand,
On the other hand, we multiply the equation with and get
We look at the difference of the two equations, and get
By the induction hypothesis, we get for the coefficients
, .
Because of
,
we get
for ,
and because of
,
we also get
.