I am working Econometrics Analysis by W.H.Greene and I am a bit stuck on some matrix calculations. I did studied Matrix algebra, but I cannot remember any rule concerning simplification or change of signe when dealing with transposed matrix. I looked a lot on the web but couldn't find anything that could help me resolve it.
For example, it is stated page 21 that the sum of square errors is : S = (y-XB)'(y-XB)
Which can be developed to obtain: y′y − B′X′y − y′XB + b′X′XB
But then, it goes directly to: y′y − 2y′XB + B′X′XB
I really don't understand how -B'X'y - y'XB = -2y'XB
I know the rule (AB)'=B'A' and the associative multiplication of matrices, but I cannot understand how the author get this result.
Also, he later differentiates y′y − 2y′XB + B′X′XB by B, and gets −2X'y + 2X'XB, a result I cannot get neither. Where do the 2 comes from in the final expression (the 2X'XB part)? And how -2y'X = -2X'y (first part of the final expression?
I hope someone will be able to help me.
Thanks in advance, Mathieu
P.S : In the text book, A' is used for transposed matrix.
$y'$, $X$, and $B$ are respectively of dimensions $(1 \times n)$, $(n \times K)$, $(K \times 1)$.
Hence, $y'Xb$ is a scalar $s$, and $s'=s$. The same applies to $b'X'y$.
Thus, $y'Xb = (y'Xb)' = (Xb)'y = b'X'y$
Yielding $-y'Xb - b'X'y = -2b'X'y$