Appendix 1

The derivation of the Gauss-Markov theorem depended upon the matrix identity,

$$(A - B C^{-1}) C (A - B C^{-1})^T - B C^{-1} B^T \equiv A C A^T - B A^T - (B A^T)^T$$ (1)

This identity is not very intuitive and so we will provide the proof here. This proof depends upon one assumption being true, that C = C^T.

Let us define,

$$X \equiv (A - B C^{-1}) C (A - B C^{-1})^T - B C^{-1} B^T$$ (2)

and,

$$Y \equiv A C A^T - B A^T - (B A^T)^T$$ (3)

If we can establish that $X \equiv Y$, then we have our proof.

We start by expanding X,

$$\equiv$$ X (A - B C^-1) C (A - B C^-1)^T - B C^-1 B^T

= (A - B C^-1) C (A^T - C^-TB^T) - B C^-1 B^T (4)

and since we have assumed that C = C^T, then C^-1 = C^-T, so

X = (A - B C^-1) C (A^T - C^-1B^T) - B C^-1 B^T

= (A C - B ) (A^T - C^-1B^T) - B C^-1 B^T

= ACA^T - BA^T - AB^T + BC^-1B^T - BC^-1B^T

$$ACA^T - BA^T - (BA^T)^T \equiv Y$$ = (5)

Q.E.D