Let be an real matrix such that
(where I denotes the n × n-identity matrix). Show that
(i) A has two eigenvalues .
(ii) Every element can be expressed uniquely as ,
where and
Take two endomorphisms of :
You have
according to the problem.
You see that
and, after a little manipulation and comaprison of dimensions,
dim Ker (u) + dim Ker (v) = n
Which is sufficient to say that
, cause it would mean v = 0
, cause it would mean u = 0
So there are effectively 2 eigenvalues for A ( 1 and -1),
and the associated eigenspaces are in direct sum (Ker u and Ker v),
which guarantees the existance and unicity if the decomposition.