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 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