A question about notation $\frac{\partial^2}{\partial\theta\partial\theta^T}l(\theta)$

Question

For $l:\mathbb{R}^n\rightarrow\mathbb{R}$ a differentiable function and $\theta$ a vector, I read this notation $\frac{\partial^2}{\partial\theta\partial\theta^T}l(\theta)$ in a paper and want to figure out what its meaning. It comes in a taylor expansion, where I expect it to be a Hesse matrix. But If $\frac{\partial}{\partial\theta}l=(\frac{\partial}{\partial\theta_1}l,...,\frac{\partial}{\partial\theta_n}l)$, $\frac{\partial}{\partial\theta^T}l=(\frac{\partial}{\partial\theta_1}l,...,\frac{\partial}{\partial\theta_n}l)^T$,this notation seems to be a transposed Hesse matrix?

Answer 1

Rigorously speaking, if $l(\theta)$ and $\partial l/ \partial \theta$ exist, and the second-order partial derivatives are continuous, on a neighborhood of some point $\theta$ of interest, then the matrix is symmetric, which means that the transposed Hessian and the Hessian are the same. So it doesn't matter how we write the transpose sign.