Nonlinear mappings preserving at least one eigenvalue

Constantin Costara; Dušan Repovš

Studia Mathematica (2010)

  • Volume: 200, Issue: 1, page 79-89
  • ISSN: 0039-3223

Abstract

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We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either F ( x ) = u x u - 1 or F ( x ) = u x t u - 1 for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying F(0) = 0 and ρ(F(x) - F(y)) = ρ(x-y) for all x and y, where ρ(·) denotes the spectral radius, then there exists γ ∈ ℂ of modulus one such that either γ - 1 F or γ - 1 F ̅ is of the above form, where F̅ is the (complex) conjugate of F.

How to cite

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Constantin Costara, and Dušan Repovš. "Nonlinear mappings preserving at least one eigenvalue." Studia Mathematica 200.1 (2010): 79-89. <http://eudml.org/doc/285482>.

@article{ConstantinCostara2010,
abstract = {We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either $F(x) = uxu^\{-1\}$ or $F(x) = ux^\{t\}u^\{-1\}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying F(0) = 0 and ρ(F(x) - F(y)) = ρ(x-y) for all x and y, where ρ(·) denotes the spectral radius, then there exists γ ∈ ℂ of modulus one such that either $γ^\{-1\}F$ or $γ^\{-1\}F̅$ is of the above form, where F̅ is the (complex) conjugate of F.},
author = {Constantin Costara, Dušan Repovš},
journal = {Studia Mathematica},
language = {eng},
number = {1},
pages = {79-89},
title = {Nonlinear mappings preserving at least one eigenvalue},
url = {http://eudml.org/doc/285482},
volume = {200},
year = {2010},
}

TY - JOUR
AU - Constantin Costara
AU - Dušan Repovš
TI - Nonlinear mappings preserving at least one eigenvalue
JO - Studia Mathematica
PY - 2010
VL - 200
IS - 1
SP - 79
EP - 89
AB - We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either $F(x) = uxu^{-1}$ or $F(x) = ux^{t}u^{-1}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying F(0) = 0 and ρ(F(x) - F(y)) = ρ(x-y) for all x and y, where ρ(·) denotes the spectral radius, then there exists γ ∈ ℂ of modulus one such that either $γ^{-1}F$ or $γ^{-1}F̅$ is of the above form, where F̅ is the (complex) conjugate of F.
LA - eng
UR - http://eudml.org/doc/285482
ER -

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