ChapterZero

Differential of the determinant

Filed under: Mathematics — Alex @ 5:14 pm 6/29/2007

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

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

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

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

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

Possibly relevant posts:

The implicit and inverse function theorems (9/12/2006)
Smooth functions (4/14/2005)
Eikonal equation (11/14/2006)

3 Comments »

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)

Comment by Anonymous — 7/3/2007 @ 5:31 am
Thanks. It’ll be interesting to see, when I get the result for det, how it simplifies to that special case.

Comment by Alex — 7/3/2007 @ 1:06 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!)

Comment by Alex — 7/5/2007 @ 10:22 pm

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Differential of the determinant

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