Today’s Theorem of the Day (TM) I used to compute the Jacobian of a radial projection. In particular, consider the map where for all . This projects all of n-space onto the surface of the unit ball, and leaves the interior untouched. Then we may compute the derivative .
To calculate the Jacobian of F means we have to calculate the determinant of that matrix. With a little figuring, we can write that last sentence as .
Now we apply The Theorem, which Terry Tao quoted Percy Deift as calling (half-jokingly) “the most important identity in mathematics”, and wikipedia calls, less impressively, “Sylvester’s determinant formula“. Its usefulness derives from turning the computation of a very large determinant into a much smaller determinant. At the extreme, we apply the formula to vectors u and v, and it says that . In our case, it yields . Thus we turned the problem of calculating the determinant of an n x n matrix into calculating the determinant of a 1 x 1 matrix.