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Linear preserver problems: A brief introduction and some special techniques. (English) Zbl 0762.15016

Let M be any one of the following matrix spaces: the set of all m×n matrices over the field 𝔽, where usually 𝔽 is or ; the set of all n×n symmetric matrices over 𝔽; the set of all n×n skew-symmetric matrices over 𝔽; the set of all Hermitian matrices. The typical linear preserving problems are:

1. let F be a (scalar-valued, vector-valued, or set-valued) function on M. Characterize those linear operators ϕ on M that satisfy F(ϕ(A))=F(A) for all AM;

2. let SM. Characterize those linear operators ϕ on M that satisfy ϕ(S)=S or S;

3. let be a relation or an equivalence relation on M. Characterize those linear operators ϕ on M that satisfy ϕ(A)ϕ(B) whenever AB (or iff AB);

4. given a transform F:MM, characterize those linear operators ϕ on M that satisfy F(ϕ(A))=ϕ(F(A)) for all AM.

This paper is a survey which gives a gentle introduction to these problems.


MSC:
15A72Vector and tensor algebra, theory of invariants
15-02Research monographs (linear algebra)
15A57Other types of matrices (MSC2000)