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" <ortiz@ups-albi.fr> 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