I understand that the trace of the projection matrix (also known as the "hat" matrix) X*Inv(X'X)*X' in linear regression is equal to the rank of X. How can we prove that from first principles, i.e. without simply asserting that the trace of a projection matrix always equals its rank?
I am aware of the post Proving: "The trace of an idempotent matrix equals the rank of the matrix", but need an integrated proof.