Differential of the determinant

June 29th, 2007 ~ Posted in: Mathematics

Recall the definition of a (Frechet) differential of a mapping between two Banach spaces B and W:

If f : U \subset B \rightarrow V \subset W, and for x \in U there is a linear mapping df_x such that

\lim_{h \rightarrow 0} \frac{\|f(x+h)-f(x)-df_x h\|}{\|h\|} = 0,

then f is said to be differentiable at x with differential df_x. If df_x is defined for all x\in U, then f is said to be differentiable with differential df.

Now assume f is differentiable and defined between an n-dimensional Banach space B and \R. Let \{e_i\}_{i=1,\ldots,n} be a basis for B, then

\lim_{h e_i \rightarrow 0} \frac{|f(x+h e_i) - f(x) - h df_x e_i|}{|h|} =0 \Rightarrow df_x e_i = \frac{\partial f}{\partial x_i} ,

so df_x(y) = y \cdot (\nabla f)(x). Likewise you can show that if f : \R^n \rightarrow \R^m, then df_x(y) = \left(\left. \frac{\partial f_i}{\partial x_j}\right|_x \right) y.

Here’s the question: let f : GL(n) \rightarrow \R be defined by f : A \rightarrow \det A. What’s df?

Mathematics ~

3 Responses to “Differential of the determinant”

  • 1. Anonymous
    July 3rd, 2007 at 5:31 am

    Here is what I remember: if you restrict the domain of f to symmetric positive definite matrices, the differential of log(det(X)) at point A is df : X -> tr(inv(A) X)

  • 2. Alex
    July 3rd, 2007 at 1:06 pm

    Thanks. It’ll be interesting to see, when I get the result for det, how it simplifies to that special case.

  • 3. Alex
    July 5th, 2007 at 10:22 pm

    Bah, I’ve moved on. Here’re some notes on Jacobi’s formula for the derivative of a determinant, with even a Taylor series expansion of the determinant (fancy!)

This entry was posted on Friday, June 29th, 2007 at 5:14 pm and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Leave a Reply