论文标题
根据真实参数的平稳避免结构化矩阵的特征值
Eigenvalue avoidance of structured matrices depending smoothly on a real parameter
论文作者
论文摘要
我们探索了避免特征值的概念，对于其他类别的结构化矩阵，这是对真实对称和遗传学矩阵的理解。 We adopt a differential geometric perspective and study the generic behaviour of the eigenvalues of regular and injective curves $t \in ]a,b[ \mapsto A(t) \in \mathcal{N} $ where $\mathcal{N}$ is a smooth real Riemannian submanifold of either $\mathbb{R}^{n \times n}$ or $ \ mathbb {c}^{n \ times n} $。我们专注于$ \ Mathcal {n} $对应于某些（真实或复杂的）结构化矩阵，包括偏斜 - 对称，偏斜，正交，统一，带带的对称，带状的Hermitian，带状的偏斜，偏斜的偏压和带式倾斜的倾斜度。我们认为，对于某些结构，避免特征值始终发生，而对于其他结构，这可能取决于大小的奇偶校验，基于多重特征值的数值，并且可能取决于确定性的价值。作为我们工具的进一步应用，我们还研究了$ \ mathbb {r}^{m \ times n} $或$ \ mathbb {c}^{m \ times n} $中的非结构化矩阵的单数值回避。
We explore the concept of eigenvalue avoidance, which is well understood for real symmetric and Hermitian matrices, for other classes of structured matrices. We adopt a differential geometric perspective and study the generic behaviour of the eigenvalues of regular and injective curves $t \in ]a,b[ \mapsto A(t) \in \mathcal{N} $ where $\mathcal{N}$ is a smooth real Riemannian submanifold of either $\mathbb{R}^{n \times n}$ or $\mathbb{C}^{n \times n}$. We focus on the case where $\mathcal{N}$ corresponds to some class of (real or complex) structured matrices including skew-symmetric, skew-Hermitian, orthogonal, unitary, banded symmetric, banded Hermitian, banded skew-symmetric, and banded skew-Hermitian. We argue that for some structures eigenvalue avoidance always happens, whereas for other structures this may depend on the parity of the size, on the numerical value of the multiple eigenvalue, and possibly on the value of determinant. As a further application of our tools we also study singular value avoidance for unstructured matrices in $\mathbb{R}^{m \times n}$ or $\mathbb{C}^{m \times n}$.
