From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Help needed: Scalar functions of a matrix
Date: 26 Oct 1996 05:25:01 GMT

In article <54gq4k$3ff@mimsy.cs.umd.edu>,
Chandra Shekhar <shekhar@cfar.umd.edu> wrote:
>I am looking for scalar functions of a matrix that
>are distributive, i.e.
>
>Given A = B C   (A,B,C are 2*2 matrices),
>I need f() satisying f(A) = f(B) f(C).

When restricted to invertible  2 x 2  matrices, this  f  will be a homomorphism
from  that group to the multiplicative group of nonzero real numbers [unless
f(I) <> 1, in which case,  f  is identically zero]. In particular,  f  will
send every matrix of determinant  1  to the number  1, since those matrices
are all in the commutator subgroup of  GL(2, R). So  f  has to be of
the form  g o det,  where  g  is a multiplcative map  R^x -> R^x. If you
assume in addition that   f  is continuous, the only choices for  g  are
powers (or powers of the absolute variable). 

(Singular matrices are not much of a problem either: if  det(A)=0, then
A^2=kA  for  k = Tr(A), so the multiplcative property would imply
f(A) = 0  or  f(A)=f( Tr(A) . I) . But the latter case would be
inconsistent with multiplication by, say  diag(r, r^(-1))  for all  r <> 0.)

So the determinant is basically your only choice. I may have missed some
subcase here but I think you must have one of these:
	f(A) = 0  for all  A
	f(A) = 1  for all  A
	f(A) = |det(A)|^s  for some real  s
	f(A) = |det(A)|^s * sign(det(A))  for some real  s
	f  not continuous.

dave