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Preservers of Eigenvalue Inclusion Sets
Herman, Aaron
Herman, Aaron
Abstract
Denote by Mn the set of n × n complex matrices. Let S(A) be one of the following: the Gershgorin Set, Braeur's Set of Cassini Ovals, and the Ostrowki set. Characterization is obtained for maps Φ : Mn → Mn such that S( Φ(A) - Φ(B)) = S(A - B) for all A,B ∈ Mn. It is shown that such maps can be decomposed as the composition of two or three simple maps such as permutation similarity transforms, permutation of off-diagonal entries in each row, and norm preserving maps on individual diagonal entries. Similar results can be obtained for maps Φ : Mn → Mn satisfying S(Φ(A)+ Φ(B)) = S(A+B) for all A,B ∈ Mn. Moreover, using these results, one can readily determine the structure of additive maps and linear maps Ψ : Mn → Mn satisfying S( Ψ(A)) = S(A) for all A ∈ Mn. Characterization is also obtained for maps Φ: Mn → Mn satifying S( Φ(A) Φ(B)) = S(AB) for all A,B ∈ Mn. In most cases, the maps are composition of similarity tranformations by permutation and diagonal unitary matrices followed by a multiplication by a scalar μ ∈ {1,-1}. Related results and problems are also described.
Description
Thesis is part of Honors ETD pilot project, 2008-2013. Migrated from Dspace in 2016.
Date
2010-05-07
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Keywords
Eigenvalue inclusion sets, Matrix preserver problem
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Department
Mathematics