From: Dave Rusin <rusin@math.niu.edu>
Date: Wed, 16 Dec 1998 07:38:06 -0600 (CST)
To: materiaux@ecole-debroglie.fr
Subject: Re:  "Hessian" matrix for f : Rn -> Rp

>Considering a function f : Rn -> Rp of n variables, the matrix of df
>(differential of f) is the jacobian matrix. Here f takes elements of Rn(the
>set of real numbers to the n) and gives vectors of Rp (not of R).
>
>If now f is a function from Rn to R, the Taylor-Young formula applied to f
>at some point a up to the second order gives three terms : the first one is
>f(a), the second one depends on the jacobian matrix of f, and the third one
>depends on the hessian matrix Q of f. Q is the matrix of a quadratic form
>the signature of which may give information about the extremums of f.
>
>My question if : what is the equivalent of this hessian matrix for f : Rn ->
>Rp ?

It has to be a quadratic map  R^n -> R^p. You should focus on the function,
not on the matrix which represents it. Indeed, the Hessian is only well-
defined after bases have been picked, while the function  f  itself and its
quadratic approximation are well-defined without reference to bases.

>Copying from the case f : Rn -> R where the hessian matrix has for general
>entry d2f/dx_idx_j, I suppose I have to define some object that could be
>represented by a cube of components (can we call this a... tensor ?) having
>d2f_k/dx_idy_j for general entry ?

Yes. You need all these data to describe the quadratic approximation to  f.
It doesn't really matter how you arrange them.

>Or do I have simply to apply the Taylor-Young formula p times, once for each
>component f_k of f ?

This is probably the most practical idea.

>Take for example the coordinate transformation from polar to cartesian :
>x = r.cos(t)
>y = r.sin(t)
>Here n=p=2. The Jacobian matrix is
>[ [cos(t)  -r.sin(t)]
>  [sin(t)   r.cos(t)]]
>What is the "hessian matrix" in this case ?

You should think about the quadratic function
	(r,t) -> (0,0) + (r, 0) + (0, rt)
if you want to approximate  f  near the origin, although since  f  has
a critical point there, you don't get much information in this particular
case. But again I stress it makes more sense to think about the function
f  and these polynomials of increasing degree which approximate it, 
rather than thinking about a bunch of matrices. Anyway, you will have no 
other choice when you start to include cubic, ... terms.

This doesn't belong in sci.math.research.

dave (moderator)