- #1

lus1450

- 40

- 1

## Homework Statement

Let ##f: M_{n \times n} \rightarrow M_{n \times n}## with ##f(X) = X^2##, where ##M_{n \times n}## denotes the vector space of ##n \times n## matrices. Show ##f## is differentiable and find its differential.

## Homework Equations

## The Attempt at a Solution

So far, I've been looking at the difference quotient in order to "guess" linear transformation ##A## that will satisfy it. We have:

$$\lim_{|h|\to 0} \frac{|f(X+h) - f(X) - Ah|}{|h|} = \lim_{|h|\to 0} \frac{|Xh + hX + h^2 - Ah|}{|h|}$$after a little simplifying.

And I was thinking for a fixed ##X##, I have ##A(h) = Xh + hX##, as I wanted to get rid of the ##Xh + hX## term in the quotient. However, I'm stuck now since it's ##Ah## and not just ##A##. I was thinking of throwing in an ##h^{-1}## to my ##A##, but that would make it non-linear. Any suggestions?