I am having trouble understanding what the following question is asking:

For the space of real n by n matrices:

Find the Frechet derivatives of A maps to A^2, and A maps to A^-2.

Anyone have any pointers?

Printable View

- September 16th 2009, 04:49 PMRelayerFinding the Frechet Derivative of a Map
I am having trouble understanding what the following question is asking:

For the space of real n by n matrices:

Find the Frechet derivatives of A maps to A^2, and A maps to A^-2.

Anyone have any pointers? - September 17th 2009, 03:10 PMJose27
Let such that then, (I'm assuming you give this space the operator norm):

and we want this to tend to zero as tends to zero, which is clearly satisfied if we define , this is clearly linear, and from which it follows that is a bounded linear operator, and as such is the derivative of at doing this for all we have a map: where .

The second one is a little trickier since the function is only defined in the subset of nonsingular matrices, and you would have to prove first that this subset is open. - September 24th 2009, 02:40 PMRebesques
J is right, but if you are looking to actually compute the Frechet derivative, try