Linear maps preserving the Lorentz-cone spectrum in certain subspaces of \(M_n\)
Let \(n \geq 3\) and consider the Lorentz cone \(\mathcal K = \{(x,x_n)\in \mathbb R^{n-1}\times \mathbb R : \lVert x \rVert \leq x_n \}\), where \(\lVert x\rVert\) is the 2-norm of \(x\). It is well-known that the Lorentz cone \(\mathcal K\) is self-dual, and so the eigenvalue complementarity problem associated with \(\mathcal K\) is, for a given real-valued \(n \times n\) matrix \(A\), finding a scalar \(\lambda \in \mathbb R\) and nonzero vector \(x \in \mathbb R^n\) such that \[x \in \mathcal K, \qquad (A-\lambda I_n)x \in \mathcal K, \qquad x^T(A-\lambda I_n)x = 0,\] where \(I_n\) denotes the \(n \times n\) matrix and \(x^T\) denotes the transpose of \(x\). The scalar \(\lambda\) is called \textit{Lorentz eigenvalue} of \(A\) and the set of all Lorentz eigenvalues of \(A\) is denoted \(\sigma_\mathcal{K}(A)\), and called the \textit{Lorentz cone spectrum of \(A\)}. The authors explore linear preserver problems related to the Lorentz cone spectrum of matrices. In other words, they obtain characterizations of linear maps \(\phi: \mathcal M \to \mathcal M\) such that \(\sigma_\mathcal{K}(\phi(A)) = \sigma_\mathcal{K}(A)\) for all \(A \in \mathcal M\), where \(\mathcal M\) is a given subspace of \(M_n(\mathbb R)\). The authors call maps of the form \(\phi(A) = PAQ\) or \(\phi(A) = PA^TQ\), where \(P\) and \(Q\) are fixed matrices, \textit{standard linear maps} (most linear preserver problems in the literature conclude that the maps have this particular form; see [\textit{C.-K. Li} and \textit{S. Pierce}, Am. Math. Mon. 108, No. 7, 591--605 (2001; Zbl 0991.15001)] for a general description). Lorentz cone spectrum preservers are considered on the following subspaces \(\mathcal M\): diagonal matrices, block-diagonal matrices of the form \(\tilde{A}\oplus[a]\) with \(a \in \mathbb{R}\) and \(\tilde{A}\in M_{n-1}(\mathbb{R})\) symmetric, block-diagonal matrices of the form \(\tilde{A}\oplus[a]\) with \(a \in \mathbb{R}\) and \(\tilde{A}\in M_{n-1}(\mathbb{R})\) arbitrary, the space of symmetric matrices, and the full matrix space \(M_n(\mathbb R)\). For the diagonal and block-diagonal spaces, the authors show (see Theorem 4.2) that Lorentz cone spectrum preservers of these spaces are, in fact, standard maps with \(P\) and \(Q\) invertible of a specified form. For the symmetric and full matrix space case, the complete description of standard linear maps that preserve the Lorentz cone spectrum are provided. It seems to be an open problem as to whether or not all Lorentz cone spectrum preservers of the symmetric and full matrix spaces must in fact be standard maps. As a byproduct of studying preservers in this paper, certain properties of the Lorentz cone spectrum (and the interplay with the usual spectrum) are developed for special classes of matrices; in particular, rank-one matrices.
scientific article; zbMATH DE number 7376337
scientific article