论文标题
线性地图保留了$ 3 \ times 3 $矩阵的Lorentz频谱
Linear maps preserving the Lorentz spectrum of $3 \times 3$ matrices
论文作者
论文摘要
对于给定的$ 3 \ times 3 $ 3 $真实矩阵$ a $，相对于Lorentz锥的特征值互补性问题包括找到一个实际数字$λ$和一个非零的矢量$ x \ in \ mathbb {r}^3 $由$ \ mathbb {r}^3 $组成的所有向量组成，形成$ 45^\ circ $或较小的角度，带有正$ z $ - 轴。我们将所有解决方案的集合$λ$都称为此特征值互补问题，为$ a $的Lorentz Spectrum。我们的工作涉及Lorentz Spectrum线性保存器在空间上的线性保存器的特征，即$ 3 $ 3 \ times 3 $ 3 $真实的矩阵，也就是说，线性地图$ ϕ：m_3 \ to m_3 $，使得lorentz spectra $ a $ a $ a $和$ ϕ（a）$相同。我们已经证明，所有此类线性保留器都采用$ ϕ（a）=（q \ oplus [1]）a（q^t \ oplus [1]）$的表格，其中$ q $是正交$ 2 \ times 2 $矩阵。
For a given $3 \times 3$ real matrix $A$, the eigenvalue complementarity problem relative to the Lorentz cone consists of finding a real number $λ$ and a nonzero vector $x \in \mathbb{R}^3$ 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 $\mathbb{R}^3$ forming a $45^\circ$ 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 $3 \times 3$ real matrices, that is, the linear maps $ϕ: M_3 \to M_3$ 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 $ϕ(A) = (Q \oplus [1])A(Q^T \oplus [1])$, where $Q$ is an orthogonal $2 \times 2$ matrix.
