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