6
$\begingroup$

Suppose I have a function $\phi(M):=M^n$ where $M\in M_n(R)$

Does it make any sense to talk about the derivative of this function? What would it mean if it does make sense? I am also wondering what the derivative (if it exists and is meaningful) would be. (Perhaps $nM^{n-1}$?)

Thanks.

  • 4
    I'd just like to say that this is a great question - keep on thinking about how ideas you've learned in di$f$ferent mathematical contexts might interact!2011-11-22

2 Answers 2

2

Yes, it makes sense to talk about derivatives of functions $f:V\to W$, where $V$ and $W$ are a normed vector space, for example.

In this case, we say that $f$ is differentiable at $x\in V$ if there is a linear mapping $Df(x):V\to V$ such that

$ \lim_{h\to 0}\frac{\|f(x+h)-f(x)-Df(x)h\|}{\|h\|}=0 $

In your particular case, you could compute the derivative of $M\mapsto M^n$ by using a $n$-linear application given by $ \varphi(A_1,\ldots,A_n)=A_1\cdot\ldots\cdot A_n. $

You can show for any $X=(X_1,\ldots,X_n)\in M_n(R)\times\ldots\times M_n(R)$ ($n$-times) that

$ D\varphi(X)(H_1,\ldots,H_n)=\sum_{i=1}^n \varphi(X_1,\ldots,X_{i-1},H_i,X_{i+1},\ldots,X_n). $

So the formula you want should be interpreted as the derivative of $\varphi$ at $X=(M,\ldots,M)$ applied at $(Id,\ldots,Id)$. Which is equivalent to what we do when we write down this formula for real and complex functions.

  • 0
    Yes, I think Frechet wrote about this originally in his paper: "Not just fresh, but Frechet"2011-11-26
1

Layman version:

Start with $n=1$: $f(M) = M$. Then $Df(M) = DM = I$.

For $n=2$: $f(M) = M*M$. Apply product rule: $Df(M) = (DM) * M + M * (DM) = I*M + M*I = 2M$

Induction on general $n\geq2$, assuming $D(M^{n-1}) = (n-1)*M^{n-2}$: $f(M) = M*M^{n-1}$. Product rule: $Df(M) = (DM) * M^{n-1} + M * D(M^{n-1}).$ Use induction: $D(M^{n-1}) = (n-1)M^{n-2}.$ Then $Df(M) = I * M^{n-1} + M*(n-1)*M^{n-2} = n*M^{n-1}.$

QED.

Of course, I'm assuming that the product rule holds.

  • 0
    No problem! I should also mention that if you see a piece of LaTeX you want to know the code for on the site, you can right click on it and choose "Show Source" - this is a good way of picking up how to do things.2011-11-22