Derivatives of (e.g.) functions between matrices

[Mathmos6]

Hi there all,

I've been doing some work to try and keep myself sharp over the holidays but I've reached a point where I'm doing maths I have no notes for, and one particular topic has me completely stuffed - a few more difficult derivatives, primarily between matrices.

http://www.dpmms.cam.ac.uk/site2002/Teaching/IB/AnalysisII/2008-2009/08sheet4.pdf

This is my last worksheet and although I managed to fumble my way through the first 2 parts of Q6, I haven't got a clue how to approach Qns 2, the rest of 6, all of 7, 8, and 10 - i.e. the majority of the questions! I can't find the definition of a derivative for these situations anywhere online, nor for 'continuously differentiable' in cases other than R^n.

However much help I can get, even things like just definitions or an example of one of these questions would be really really appreciated, this is the only thing I can't seem to get moving on in all my work so the more assistance I can get the better.

Thankyou -very- much in advance!

 

[Mark44]

Hi there all,

I've been doing some work to try and keep myself sharp over the holidays but I've reached a point where I'm doing maths I have no notes for, and one particular topic has me completely stuffed - a few more difficult derivatives, primarily between matrices.

http://www.dpmms.cam.ac.uk/site2002/Teaching/IB/AnalysisII/2008-2009/08sheet4.pdf

This is my last worksheet and although I managed to fumble my way through the first 2 parts of Q6, I haven't got a clue how to approach Qns 2, the rest of 6, all of 7, 8, and 10 - i.e. the majority of the questions! I can't find the definition of a derivative for these situations anywhere online, nor for 'continuously differentiable' in cases other than R^n.

However much help I can get, even things like just definitions or an example of one of these questions would be really really appreciated, this is the only thing I can't seem to get moving on in all my work so the more assistance I can get the better.

Thankyou -very- much in advance!

For 2, the problem is to show that, for A $\in$ M_n, and f(A) = A⁴, that f is differentiable, and to find a formula for Df |_A.

The usual way to find a derivative is to evaluate the limit of the difference quotient. That is, to evaluate this limit:
$$\lim_{H \rightarrow 0}H^{-1}[f(A + H) - f(A)]$$

The obvious differences between this limit and the form you've no doubt seen before are 1) 0 is the nxn zero matrix, and 2) instead of dividing by h, we're multiplying by H^-1.

This same tactic might be of help to you in #6 as well.