# Appendix 1

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

 (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 = CT.

Let us define,

 (2)

and,

 (3)

If we can establish that , then we have our proof.

We start by expanding X,

 X (A - B C-1) C (A - B C-1)T - B C-1 BT = (A - B C-1) C (AT - C-TBT) - B C-1 BT (4)

and since we have assumed that C = CT, then C-1 = C-T, so
 X = (A - B C-1) C (AT - C-1BT) - B C-1 BT = (A C - B ) (AT - C-1BT) - B C-1 BT = ACAT - BAT - ABT + BC-1BT - BC-1BT = (5)

Q.E.D

Skip Carter
1999-12-09