Another nice theorem

Trying to visualize the projection map using fibers. You'll have to take my word that the lines stop before getting to the origin.

Today’s Theorem of the Day (TM) I used to compute the Jacobian of a radial projection.  In particular, consider the map F: \mathbb{R}^n \to \mathbb{R}^n where x \mapsto x/|x| for all |x| > 1.  This projects all of n-space onto the surface of the unit ball, and leaves the interior untouched.  Then we may compute the derivative \frac{\partial F_j}{\partial x_k} = \frac{\delta_{j,k}|x|^2 - x_k^2}{|x|^3}.

To calculate the Jacobian of means we have to calculate the determinant of that matrix.  With a little figuring, we can write that last sentence as |JF(x)| = \det \left(\frac{1}{|x|} ( I - \frac{x^Tx}{|x|^2} ) \right) = \frac{1}{|x|^n} \det \left(I-\frac{x^Tx}{|x|^2}\right).

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 \det (I+u^Tv) = 1+v^Tu.  In our case, it yields |JF(x)| = 0.  Thus we turned the problem of calculating the determinant of an n x n matrix into calculating the determinant of a 1 1 matrix.

Pretty nifty.


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