From: wcw@math.psu.edu (William C Waterhouse)
Subject: Re: interesting linear algebra question
Date: 25 Feb 1999 20:43:25 GMT
Newsgroups: sci.math
Keywords: Determinant-preserving endomorphisms of End(V)
In article <01be5fd2$0542fe40$37b939c2@buromath.ups-albi.fr>,
"pascal ORTIZ" writes:
> Let K be a field.
>
> Determine the truth or the falsity of the following statement:
>
> "The only K-linear transformation f:M_n(K) ---> M_n(K)
> determinant-preserving, namely
>
> det(M)=det(f(M)) for all M in M_(K)
>
> are of type f(M)=UMV or of type f(M)=U(M*)V
> where U and V are given matrices such that det(UV)=1
> and M* is the transpose of the matrix M."
>....
This theorem was discovered by Frobenius
(Abhandlungen III, 83-103) and has been rediscovered many
times in different contexts. There is a paper of mine,
"Automorphisms of det(X_{ij}): The group scheme approach"
(Advances in Mathematics 65 (1987) 171-203), which discusses
the form the result takes over commutative rings; it
contains references to several other treatments over
fields.
There are many papers on problems of this type. I remember an issue
of Linear and Multilinear Algebra devoted to the topic, and
I see that MathSciNet turns up the following references:
Linear preserver problems references. A survey of linear
preserver problems. Linear and Multilinear Algebra 33 (1992),
no. 1-2, 121--129.
Beasley, LeRoy; Li, Chi-Kwong; Pierce, Stephen Miscellaneous
preserver problems. A survey of linear preserver problems.
Linear and Multilinear Algebra 33 (1992), no. 1-2, 109--119.
William C. Waterhouse
Penn State