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Linear preserver problems. (English) Zbl 0991.15001
Let $M_{m,n}$ denote the algebra of all complex $m\times n$ matrices, and put $M_{n}:=M_{n,n}$. A “linear preserver problem” is a problem in which we want to describe the set of all linear operators on $M_{m,n}$ which map a prescribed subset $S$ of the algebra into itself. An early example by {\it J. Dieudonné} [Arch. Math., Oberwolfach 1, 282-287 (1949; Zbl 0032.10601)] settled the case where $S$ is the set of all singular matrices. In this case, for every invertible linear operator $\phi$ of $M_{n}$ preserving $S$ there exist invertible matrices $M$ and $N$ such that $\phi$ has one of the forms: (1) $\phi(A)=MAN$ for all $A\in M_{n}$; or (2) $\phi(A)=MA^{t}N$ for all $A\in M_{n}$. There is now an extensive literature on linear preserver problems. Part of the interest lies in the fact that for many naturally defined subsets the linear preservers are always of the forms (1) and (2) (sometimes with extra conditions on $M$ and $N$) or are simple variants of these forms. This paper describes a variety of questions which have been considered and the techniques used (which include, in addition to linear algebra, algebraic geometry, Lie methods, differential geometry and model theory). It surveys some of the extensive literature and points to open problems.

##### MSC:
 15A04 Linear transformations, semilinear transformations (linear algebra)
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