Vector notation unclear step

Question

In my econometrics textbook, I have this step which is not clear to me:

\begin{align} S &= e'e \\ &= (y-W\beta)'(y-W\beta) \\ &= \underbrace{y'}_{1\times T} \ \underbrace{y}_{T\times 1} -\underbrace{y'}_{1\times T} \ \underbrace{W\beta}_{(T\times2)(2\times1)} -\underbrace{\beta'W'y}_{(1\times2)(2\times T)(T\times1)} +\underbrace{\beta'W'W\beta}_{1\times1}, \end{align} or, since all the terms are scalars, $$S =y'y -2\beta'W'y +\beta'W'W\beta. \tag{2.5.6}$$

Basically, I don't understand how we go from $-y'W\beta -\beta'W'y$ to $-2\beta'W'y'$. Thanks in advance.

Answer 1

$y^\prime W \beta$ is a 1 by 1 matrix (a scalar). As such, it is equivalent to its own transpose (if you exchange rows and columns in a matrix with one row and one column, you have changed nothing).

So $$y^\prime W \beta = (y^\prime W \beta)^\prime = \beta^\prime W^\prime y$$

Answer 2

Why is $y′\ W\beta$ a $1$ by $1$ matrix?

If you want to evaluate the dimensions of a product you just have to look at the dimensions of the very left factor and the very right factor. The number of rows of the product is the number of the rows of the very left factor ($\color{blue}{\text{blue}}$). And the number of columns of the product is the number of the columns of the very right factor ($\color{red}{\text{red}}$)

$$\large{\underbrace{\beta'W'y}_{(\color{blue}1\times \color{green}2)(\color{green} 2\times \color{pink} T)( \color{pink} T\times \color{red}1)}} $$

The other dimensions (green, pink) are cancelling out each other.