Projection Matrix Consider linear regression problem: min . Ax - y B (2) we say the optimal solution is x* = (A"A) ATy, which states that any y projected to space A is given by y = Ax* =A(ATA)-ATy, and we define A(ATA) AT as the projection matrix P. . Please verify P is symmetric . Show that if each column of A is linearly independent, then A"A is positive definite and thus invertible . P = P for any n which is positive integer . Eigenvalue of P is either 0 or 1 . trace(P) = rank(P) After you have proved all the above, please use Python/Matlab to randomly generate A (more rows than columns, say A ( R10x5) and manually verify the correctness of the above conclusions