A result from linear algebra states that r(AB) lessthanorequalto min{r(y4), tau(

Question

A result from linear algebra states that r(AB) lessthanorequalto min{r(y4), tau(5)} for any matrices A and B of appropriate order. (The number of columns of A must match the number of rows of B so that the product AB can be formed.) You may wish to use this result to help you prove the following. (a) For any matrix X, r(X) = r(X'X). (b) For any matrix X, tau(X) = r(Px). (c) For the linear model y = X beta + e, suppose that H beta is estimable with full row rank. Show that the rank of H(X'X)^-H' is k, where k is the number of rows of H. Note that H(X'X)^-H' is a k x k matrix. Px = X (X'X) X'

Explanation / Answer

Please find the solution of problem below:

Since we are given that Rank(AB)<= min{Rank(A),Rank(B)}

==> Rank(AB)<= Rank(A) and Rank(AB)<= Rank(B) -------------------(I)

Now we need to prove that Rank(X) = Rank(Px)

==> Rank(Px) = Rank(X(X'X)-X) Since Px = X(X'X)-X

==> Ran((Px) = Rank(X(X'X)-X) <= Rank(X) Using the result of above equation ---(I)

==>Ran((Px) = Rank(X(X'X)-X) <= Rank(X) = Rank(PxX)

==> Rank(Px) = Rank(X(X'X)-X) <= Rank(X) = Rank(PxX)<=Rank(Px) ------(II)

As Px is Idempotent Matrix & from the equations (I) & (II) we have   Rank(X) = Rank(Px)

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