Chapter 30

# Linear Preserver Problems

Peter Šemrl

University of Ljubljana

Linear preservers are linear maps on linear spaces of matrices that leave certain subsets, properties, relations, functions, etc. invariant. Linear preserver problems ask what is the general form of such maps. Describing the structure of such maps often gives a deeper understanding of the matrix sets, functions, or relations under the consideration. Some of the linear preserver problems are motivated by applications (system theory, quantum mechanics, etc.).

## 30.1 Basic Concepts

Definitions:

Let V be a linear subspace of Fm×n. Let f be a (scalar-valued, vector-valued, or set-valued) function on V, M a subset of V, and ~ a relation defined on V.

A linear map φ : V → V is called ...

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