The method of matrix deformations
by
Alexander Guterman
The Moscow State University, Russia

The classification of linear transformations under which certain matrix properties are invariant was the object of intensive study during the past century. A detailed and self-contained survey of linear preservers for different functions, properties, and relations can be found in [1]. In many cases the approach to classify linear transformations preserving matrix invariants is to use the reductions that are based on Galois Connections. Namely, researchers switch from one context A to another context B which is more known and better understood. However there were no systematic and efficient methods to do the reductions.

We present the method of Matrix Deformations which unifies the reduction technique mentioned above to solve Linear Preserver Problems. Our method alowes one to replace a given set or relation with its deformation such that linear preservers of the deformation may be found easily and then to apply this for the classification of the linear preservers for original set or relation. This method is applicable for many different Linear Preserver Problems and allowes us to prove several new classification theorems.

[1] S. Pierce and others, A Survey of Linear Preserver Problems, Linear and Multilinear Algebra, 33 (1992), 1-119.

Date received: January 28, 2001