论文标题
Hermitian矩阵的光谱传播
The spectral spread of Hermitian matrices
论文作者
论文摘要
令$ a $为a $ n \ times n $复杂的Hermitian矩阵，让$λ（a）=（λ_1，\ ldots，λ_n）\ in \ mathbb {r}^n $表示$ a $ a $ a $ a $ a的特征，计数多重性并以非进度顺序排列。由低等级矩阵近似理论引起的问题，我们研究$ a $的光谱传播，表示为$ \ text {spr}^+（a）$，由$ \ text {spr} {spr}^+（a）=（λ_1_1-1-λ_{n} λ_{k}-λ_{n-k+1}）\ in \ mathbb {r}^k $，其中$ k = [n/2] $（integer part）。频谱差是$ a $频谱分散的矢量值衡量标准，它允许人们获得几种次劳动不等式。在目前的工作中，我们获得了与陶的不平等相关的不平等性，这些不平等是针对正半纤维矩阵的抗diagonal，Zhan的不平等，Zhan的不平等现象的不平等现象的不平等现象是半际矩阵差异的奇异价值，超极性矩阵的极端差异，直接旋转的极端旋转，在子公积之间，全身型的矩阵之间，全身矩阵之间的矩阵和距离之间的矩阵中的矩阵中的单位或单位单位或单位单位型的矩阵。
Let $A$ be a $n\times n$ complex Hermitian matrix and let $λ(A)=(λ_1,\ldots,λ_n)\in \mathbb{R}^n$ denote the eigenvalues of $A$, counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of $A$, denoted $\text{Spr}^+(A)$, given by $\text{Spr}^+(A) =(λ_1-λ_{n}\, , \, λ_2-λ_{n-1},\ldots, λ_{k}-λ_{n-k+1})\in \mathbb{R}^k$, where $k=[n/2]$ (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of $A$, that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.
