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Abstract
For a given
3
× 3
real matrix
A ,
the eigenvalue complementarity problem relative to the Lorentz cone consists of finding a real
number
λ and a
nonzero vector x
∈ ℝ 3
such that
x T ( A
−
λ I ) x
= 0
and both
x
and
( A
−
λ I ) x
lie in the Lorentz cone, which consists of all vectors in
ℝ 3 forming a
4 5 ∘ or smaller angle with the
positive z -axis. We refer
to the set of all solutions
λ
to this eigenvalue complementarity problem as the Lorentz spectrum of
A . Our
work concerns the characterization of the linear preservers of the Lorentz spectrum on the
space
M 3 of
3
× 3 real matrices, that is, the
linear maps
ϕ
: M 3
→ M 3 such that
the Lorentz spectra of
A
and
ϕ ( A ) are the
same for all
A .
We have proven that all such linear preservers take the form
ϕ ( A )
=
( Q
⊕
[ 1 ] ) A ( Q T
⊕
[ 1 ] ) , where
Q is an orthogonal
2
× 2 matrix.
Keywords
Lorentz cone, Lorentz eigenvalues, linear preservers, $3
\times 3$ matrices
Mathematical Subject Classification
Primary: 15A18, 58C40
Milestones
Received: 31 August 2022
Accepted: 28 January 2023
Published: 15 March 2024
Communicated by Stephan Garcia
© 2024 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers).