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.