I am dealing with an example to show that the matrix($M = I − X(X'X)^{−1}X'$) is idempotent. X is a matrix with T rows and k columns and I the unit matrix of dimension T. And then to determine the rank of this matrix by using the properties of the trace of the matrix.
1.
Idempotent means that matrix $A^2=A*A=A$
$$M = I − X(X'X)^{−1}X'$$
$$M = XX' − X(X'X)^{−1}X'$$
$$MM = (XX' − X(X'X)^{−1}X')(XX' − X(X'X)^{−1}X')$$
$$MM = (XX' − X(X'X)^{−1}X')(I − I)$$
-> MM is not idempotent
Is that correct?
2.
$$Tr(AB)=Tr(BA)$$
$$Tr(M)=Tr(I − X(X'X)^{−1}X') = Tr(I − I) = Tr(0) = 0$$
Are my assumptions correct?
UPDATE
$$Tr(A-B)=Tr(A) – Tr(B)$$
$$Tr(M)=Tr(I) − Tr(X(X'X)^{−1}X')) = Tr(I) − Tr(I) = Tr(0) or Tr(I)= Rank(n)$$
Is this correct?
Best Answer