Matrix Multiplication in quadratic form and double product

Question

Given a quadratic form:

$$(FW+b)^T M (FW+b)$$

what i got is: $$W^TF^TMFW+ W^TF^TMb+ b^TMFW + b^TMb$$

my professors sums the central terms in :

$$ 2W^TF^TMb $$ but how is this possible aren't they different?

Answer 1

For typing convenience, let $\,a=FW,\:$ then the expansion becomes $$\eqalign{ \def\c#1{\color{red}{#1}} (a+b)^TM(a+b) &= a^TMa + \c{a^TMb + b^TMa} + b^TMb \\ }$$ Focus on the terms in red.

A scalar value can be thought of as a $(1\times 1)$ matrix, so it's technically symmetric.
The matrix $M$ is also symmetric, therefore $${a^TMb = (a^TMb)^T = b^TM^Ta = b^TMa}$$