Derivative of Determinant Map

Solution 1:

Write $U((V_1,V_2,W_1,W_2)) = V_1 W_2-V_2 W_1$. Then just compute the partial derivatives: $\frac{\partial U(V_1,V_2,W_1,W_2)}{\partial V_1} = W_2$, $\frac{\partial U(V_1,V_2,W_1,W_2)}{\partial V_2} = -W_1$, $\frac{\partial U(V_1,V_2,W_1,W_2)}{\partial W_1} = - V_2$, $\frac{\partial U(V_1,V_2,W_1,W_2)}{\partial W_2} = V_1$. Then the derivative at $(V,W)$ in the direction $(H,K)$ is given by $$DU((V,W))((H,K)) = W_2 H_1-W_1 H_2-V_2 K_1+V_1 K_2 = \det(H,W)+\det(V,K)$$

This can also be written in terms of the Frobenius inner product as

$$DU((V,W))((H,K)) =\langle \begin{bmatrix} W_2 & -V_2 \\ -W_1 & V_1\end{bmatrix}, \begin{bmatrix} H_1 & K_1 \\ H_2 & K_2 \end{bmatrix} \rangle$$

and so we can write the gradient $$ \nabla U((V,W)) = \begin{bmatrix} W_2 & -V_2 \\ -W_1 & V_1\end{bmatrix}$$