square of a number (vector times a matrix)

Question

This is something that is bothering me and I can't seem to remember the answer. Say you have a vector, $\alpha$ that is $p \times 1$, and a vector $X$ that is the same dimensions (assume all entries are real). Then clearly $\alpha^{\prime}X$ is just a scalar. If I write $(\alpha^{\prime}X)^2$ that is just a square of a scalar. The question is how does one expand this: I would write $\alpha^{\prime}X\alpha^{\prime}X$ but equally reasonably one can write $\alpha^{\prime}XX^{\prime}\alpha$ since the transpose of a scalar is just that scalar. Is there a reason to prefer one over the other?

Answer 1

By associativity, $$ (a'X)(X'a)=a'(XX')a $$ which is a quadratic form.

Answer 2

No, there is no reason to prefer one over the other. As you mentioned the transpose of a scalar is the scalar. In the vector notation the following holds: $(\alpha'X)' =X'\alpha$. The same goes for matrices.