In this paper, we completely characterize the linear maps \(\phi :\mathcal {M} \rightarrow \mathcal {M}\) that preserve the Lorentz-cone spectrum, when \(\mathcal {M}\) is one of the following subspaces of the space \(M_{n}\) of \(n\times n\) real matrices: the subspace of diagonal matrices, the subspace of block-diagonal matrices \(\widetilde{A}\oplus [a]\), where \(\widetilde{A}\in M_{n-1}\) is symmetric, and the subspace of block-diagonal matrices \(\widetilde{A}\oplus [a]\), where \(\widetilde{A}\in M_{n-1}\) is a generic matrix. In particular, we show that \(\phi\) should be what we call a standard map, namely, a map of the form \(\phi (A)=PAQ\) for all \(A\in \mathcal {M}\) or \(\phi (A)=PA^{T}Q\) for all \(A\in \mathcal {M},\) for some matrices \(P,Q\in M_{n}\). We then characterize the standard maps preserving the Lorentz-cone spectrum, when \(\mathcal {M}\) is the subspace \(S_{n}\) of symmetric matrices. The case \(\mathcal {M=}M_{n}\) was considered in a recent paper by Seeger (LAA 2020). We include it here for completeness.

在本文中，我们完全刻画了保持洛伦兹锥谱的线性映射\(\phi :\mathcal {M} \rightarrow \mathcal {M}\)，当\(\mathcal {M}\)是其中之一时的空间的以下子空间\（M_ {N} \）的\（N \ n次\）实矩阵：对角矩阵的子空间，块对角矩阵的子空间\（\ widetilde {A} \ oplus [一]\)，其中\(\widetilde{A}\in M_{n-1}\)是对称的，块对角矩阵的子空间\(\widetilde{A}\oplus [a]\)，其中\ (\widetilde{A}\in M_{n-1}\)是一个通用矩阵。特别地，我们证明\(\phi\)应该是我们称之为标准地图，即，地图上的形式的\（\披（A）= PAQ \）对所有\（A \在\ mathcal {M} \）或\（\披（A）= PA^{T}Q\)用于所有\(A\in \mathcal {M},\)对于某些矩阵\(P,Q\in M_{n}\)。然后，当\(\mathcal {M}\)是对称矩阵的子空间\(S_{n}\) 时，我们表征保留洛伦兹锥谱的标准映射。Seeger (LAA 2020) 最近的一篇论文中考虑了这种情况\(\mathcal {M=}M_{n}\ )。为了完整起见，我们将其包含在此处。