In this video, it is shown that Jacobian Matrices of entities such as an LU Decomposition or an Eigenproblem can be computed. However, relative to, i.e., the matrix squaring function, it is less clear how rows and columns should be ordered, not in the sense of getting the Jacobian definition right, but of designating the input variables as first input, second input, etc., and the output varibles as first output, second output, etc., which will lead to different Jacobians. Indeed, when I replicated the linked Jacobian of the LU decomposition in Mathematica, the first two rows were swapped relative to in Julia.
I compared the nonlinear transformations for a simplified $2$-vector example here. If you want to replicate what I did, replace Cell $1$ with $X(s, t) = s + \sin t$ and Cell $2$ with $Y(s, t) = t + \sin s$ then press play on Cell $22$. Then repeat this for $X(s, t) = t + \sin s$, $Y(s, t) = s + \sin t$ and play the animations side-by-side with synced $R$ values. The result seems to be that the latter is globally the same nonlinear transformation as the former, except with an extra reflection about $y = x$ integrated. Reordering the inputs instead of the outputs leads to exactly the same change, and reordering both the inputs and the outputs leads to the original transformation (this is just switching the symbols $s$ and $t$).
In some contexts $s$ and $t$ may be stipulated leading to a unique Jacobian, but in contexts like LU Decomposition or Eigenproblems, we're starting from effectively unordered sets of input variables and output variables. If there is actually an equivalence class of Jacobians in such cases rather than a single one, then perhaps it is natural to call the Jacobians without the extra reflection, or (equivalently?) the Jacobians inducing the least aggregate displacement of points, "reduced."
How does this hypothesis hold up when generalized to high-dimensional Jacobians of matrices which we can't actually visualize? Is it still the case that any reordering of inputs and outputs leads to the same transformation up to synthesis with a number of reflections about lines diagonal to axes, and is it right to view these reflections as unreduced clutter obscuring the essence of the transformation without adding any additional information (at least to an alien who could actually view the high-dimensional transformation)? Does this hypothesis agree or conflict with any others which may exist for ordering inputs and outputs, either generically (I'm not aware of any), or common yet contextual (such as vec'ing the input matrix and output matrix to get the Jacobian of the matrix squaring function, or at least ensuring that the input order and output order match in terms of matrix element positions)? Is there a way to actually find the reduced Jacobian (ideally at the time of ordering inputs and outputs but if necessary after some computations)?
