Proving a jacobian matrix is symmetric

Question

I am following this tutorial and in it they make a statement that the jacobian is symmetric. So, I have $\partial f(\mathbf{x}) / \partial {x}$, and I want to mathematically prove that it is symmetric. Here, $f(\mathbf{x})$ is softmax function.

I know that $f(\mathbf{x}) = f(\mathbf{x})^T$ must be satisfied to prove that two matrices are symmetric. However, I'm not sure how to mathematically begin doing this. Should I be re-deriving softmax in a different way to incorporate the "transposed" portion of it?

Answer 1

The off-diagonal $i,j$ entries of the Jacobian in this specific case are $$ -s_i(\mathbf x)s_j(\mathbf x) $$ as mentioned in the paragraph just before where they state that the Jacobian is symmetric. This is clearly equal to $$ -s_j(\mathbf x)s_i(\mathbf x) $$ showing that the Jacobian in this specific case is symmetric.

In general, Jacobian matrices (even square ones) are not symmetric.