I've been working on a Econometrics homework problem, and I've gotten stuck. I've completed all of it except for calculating the eigen values. The question is:
Let X be a (T x k) matrix whose columns are linearly independent, and let M = IT - X(X'X)^-1X', where IT is a (T x T) identity matrix. Show that M is symmetric and idempotent. Calculate the eigenvalues and the trace of M.
I've shown that it is symmetric and idempotent, and I got (T - k) for the trace. Without knowing the specific dimensions of the matrix, how do you calculate the eigenvalues of the matrix? It doesn't seem like there is enough information to solve the problem (I'm sure there is, but I just don't know how to piece it together). Anyways, any help you could offer with this problem would be appreciated.
I think that Seber's book is the best for this stuff.
I loved that book in grad school.
He proves in the appendix that the trace of P equals the rank of P,
where P is your projection matrix ...
I will use c for lambda.
Px=cx, where c is our e.value and x our e.vector.
Then multiply on the left by x', the transpose of x.
cx'x=x'Px=x'P^2x by idempotent
SO all our e.values are 0 or 1.
P has r e.values equal to 1 and rest equal to 0.
WHERE r is the rank.
Note that if P is symmetric and impotent (lol)
then so is I-P.