We have learned in our earlier article on Block Designs, about the C-matrix and the D-matrix. Today, I will discuss some properties of these matrices.
Result 1: C is determined entirely by the incidence matrix N.
Result 2: C is symmetric, non-negative definite, and positive semi-definite with at least one eigen value equal to 0.
Result 3: The reduced normal equations are consistent irrespective of the rank of C.
Proof: We can easily see that:
Now, , which implies Q belongs in the column space of C, completing the proof.
Result 4: , under .
Result 5: under .
Result 6: Rank(X’X)=b+Rank(C).
Since rank remains unchanged by pre- or post- multiplication by a non singular matrix, consider UAU’ which will have same rank as A.
, Thus Proved.
Properties of D-matrix:
- D is symmetric and .
So we can see: b+rank(C)=v+rank(D).
Thus for a block design error degrees of freedom is (n-rank(X)) or (n-b+rank(C)).