The C-matrix

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:

C=X_{\tau}'[I-P_{X_{\beta}}]X_{\tau}=(X_{\tau}'P_{X_\beta}^{^{\perp}})'(X_{\tau}'P_{X_\beta}^{^{\perp}}) \Rightarrow \textrm{Column Space of C}=\textrm{Column Space of} X_{\tau}'P_{X_\beta}^{^{\perp}}

Now, Q=T-NK^{-1}B=(X_{\tau}'P_{X_{\beta}}^{^{\perp}})Y, which implies Q belongs in the column space of C, completing the proof.

Result 4: E(Q)=C\tau , under \Omega.

Result 5: D(Q)=C\sigma^2  under \Omega.

Result 6: Rank(X’X)=b+Rank(C).

Proof:    Rank(X'X)=Rank\begin{pmatrix} R & N \\ N' & K \end{pmatrix}

Let     U=\begin{pmatrix} I_v & -NK ^{-1} \\ 0 & I_b \end{pmatrix}

Since rank remains unchanged by pre- or post- multiplication by a non singular matrix, consider UAU’ which will have same rank as A.

Rank(A)=Rank(UAU')=b+Rank(C), Thus Proved.

Properties of D-matrix:

  • D is symmetric  and D1_b=0.
  • E(P)=D\beta
  • D(P)=D\sigma^2
  • Rank(X'X)=v+Rank(D)

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)).

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