Linear maps preserving the Lorentz spectrum of 3 × 3 matrices

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 -λI)x = 0 and both x and (A -λI)x lie in the Lorentz cone, which is comprised of all vectors in R 3 forming a 45 • 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), where Q is an orthogonal 2 × 2 matrix.