I'm looking for some help on a linear algebra problem that ultimately can be simplified to as follows. Thanks in advance!
Let $J$ be an arbitrary full-rank 3 by 3 matrix and $n$ an arbitrary unit 3-d vector, denote $A=J[n]_\times$ where $[n]_\times=\begin{bmatrix}0&-n_3&n_2\\n_3&0&-n_1\\-n_2&n_1&0\end{bmatrix}$ is the skew-symmetric matrix representation of $n$.
Now because $rank(A)=2$, $A$ only has two non-zero singular values $\sigma_1,\sigma_2$. I'm wondering if there is a closed form expression for the squared difference between these two singular values, i.e. $d=(\sigma_1-\sigma_2)^2$ w.r.t. $J$ and $n$?
(I tired to solve this using symbolic algebra in python (e.g. SymPY), but it gave me a long expression that's not really helpful. I need a compact expression that bypasses SVD and can be easily implemented in an automatic differentiaion framework.)