论文标题
Finsler的几何形状严格增生矩阵
Finsler geometries on strictly accretive matrices
论文作者
论文摘要
在这项工作中，我们研究了一组严格的积聚矩阵，即具有正定遗传部分的矩阵集，并表明该集可以将其解释为平滑的歧管。使用最近提出的对称极性分解对部门矩阵，我们表明该歧管与（Hermitian）阳性确定矩阵的歧管直接产物和严格的积聚单位矩阵的歧管是不同的。利用这种分解，我们介绍了一个芬斯勒指标家族，并将其用于大地测量和地球距离的特征。最后，我们将大地距离应用于矩阵近似问题，还对引入的几何形状与严格增生矩阵的几何均值之间的关系发表了一些评论，这些矩阵由S. drury在[S. Drury，线性多线性代数。 2015 63（2）：296-301]。
In this work we study the set of strictly accretive matrices, that is, the set of matrices with positive definite Hermitian part, and show that the set can be interpreted as a smooth manifold. Using the recently proposed symmetric polar decomposition for sectorial matrices, we show that this manifold is diffeomorphic to a direct product of the manifold of (Hermitian) positive definite matrices and the manifold of strictly accretive unitary matrices. Utilizing this decomposition, we introduce a family of Finsler metrics on the manifold and charaterize their geodesics and geodesic distance. Finally, we apply the geodesic distance to a matrix approximation problem, and also give some comments on the relation between the introduced geometry and the geometric mean for strictly accretive matrices as defined by S. Drury in [S. Drury, Linear Multilinear Algebra. 2015 63(2):296-301].
