In my statistics class, I was introduced to Fisher Information. As it comes from the Taylor Expansion in vector form, I wanted to know terms were ordered in a certain way - whether it was just to make it work or there was some intuition behind it. Taking the general Taylor Expansion as an example,
$f(x)=f(x_0)+\nabla f(x_0)(x-x_0)+\frac{1}{2}(x-x_0)^T\nabla\nabla^Tf(x_0)(x-x_0).$
I am mainly fussed with the quadratic term. Would I be correct in saying that $y^Ty$ is like squaring a number? And if so, what is the meaning and reasoning behind sticking the derivative between these two terms - why cannot it be interchanged with the first transpose.