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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 $ℝ$ or $ℂ$; 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 $\varphi$ on $M$ that satisfy $F\left(\varphi \left(A\right)\right)=F\left(A\right)$ for all $A\in M$;

2. let $S\subset M$. Characterize those linear operators $\varphi$ on $M$ that satisfy $\varphi \left(S\right)=S$ or $\subset S$;

3. let $\sim$ be a relation or an equivalence relation on $M$. Characterize those linear operators $\varphi$ on $M$ that satisfy $\varphi \left(A\right)\sim \varphi \left(B\right)$ whenever $A\sim B$ (or iff $A\sim B\right)$;

4. given a transform $F:M\to M$, characterize those linear operators $\varphi$ on $M$ that satisfy $F\left(\varphi \left(A\right)\right)=\varphi \left(F\left(A\right)\right)$ for all $A\in M$.

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

##### MSC:
 15A72 Vector and tensor algebra, theory of invariants 15-02 Research monographs (linear algebra) 15A57 Other types of matrices (MSC2000)