## 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 $$m\times n$$ matrices over the field $$\mathbb{F}$$, where usually $$\mathbb{F}$$ is $$\mathbb{R}$$ or $$\mathbb{C}$$; the set of all $$n\times n$$ symmetric matrices over $$\mathbb{F}$$; the set of all $$n\times n$$ skew-symmetric matrices over $$\mathbb{F}$$; 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 $$\varphi$$ on $$M$$ that satisfy $$F(\varphi(A))=F(A)$$ for all $$A\in M$$;
2. let $$S\subset M$$. Characterize those linear operators $$\varphi$$ on $$M$$ that satisfy $$\varphi(S)=S$$ or $$\subset S$$;
3. let $$\sim$$ be a relation or an equivalence relation on $$M$$. Characterize those linear operators $$\varphi$$ on $$M$$ that satisfy $$\varphi(A)\sim\varphi(B)$$ whenever $$A\sim B$$ (or iff $$A\sim B)$$;
4. given a transform $$F:M\to M$$, characterize those linear operators $$\varphi$$ on $$M$$ that satisfy $$F(\varphi(A))=\varphi(F(A))$$ for all $$A\in M$$.
This paper is a survey which gives a gentle introduction to these problems.
Reviewer: V.L.Popov (Moskva)

### MSC:

 15A72 Vector and tensor algebra, theory of invariants 15-02 Research exposition (monographs, survey articles) pertaining to linear algebra 15B57 Hermitian, skew-Hermitian, and related matrices
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