关于矩阵的Schatten p-范数的注记  

Remarks on the Schatten p-norm of matrices

作　　者：任芳国[1] 和嘉琪 REN Fang-guo;HE Jia-qi(School of Mathematics and Statistics,Shaanxi Normal University,Xi an 710119,China)

机构地区：[1]陕西师范大学数学与统计学院,陕西西安710119

出　　处：《东北师大学报(自然科学版)》2023年第4期1-8,共8页Journal of Northeast Normal University(Natural Science Edition)

基　　金：国家自然科学基金资助项目(11471200).

摘　　要：利用矩阵奇异值分解、柯西不等式及Schatten p-范数的酉不变性,讨论了矩阵主对角线元素与矩阵Schatten p-范数之间的关系.利用正交投影的性质及分块矩阵的主对角块组成的准对角矩阵可以表示成其凸组合,刻画了分块矩阵与其主对角块p-范数之间的关系.利用分块矩阵的技巧、矩阵的谱分解及Schatten p-范数的特性,深入讨论了矩阵与其伴随换位子Schatten p-范数之间的关系.利用了正规矩阵的特性及Frobenius范数的特性,给出了矩阵的绝对值及换位子之间Frobenius范数的界.所得结果细化和深化的矩阵Schatten p-范数的已有结果.Matrix norm is an important numerical index to measure the size of matrix and plays an important role in quantum information.The relationship between the main diagonal elements of the matrix and the matrix norm is deeply discussed by using the singular value decomposition of the matrix,Cauchy inequality and the unitary invariance of the Schatten p-norm.The relationship between the block matrix and its main diagonal block p-norm is characterized by using the properties of orthogonal projection and the quasi diagonal matrix composed of the main diagonal blocks of the block matrix,Then,the relationship between the matrix and its adjoint commutator Schatten p-norm is deeply discussed by using the technique of block matrix,the spectral decomposition of the matrix and the characteristics of Schatten p-norm.The absolute value of the matrix and the bound of Frobenius norm between commutators are given by using the characteristics of normal matrix and Frobenius norm.The obtained results refine and deepen the existing results of the matrix Schatten p-norm.

关 键 词：Schatten p-范数 奇异值分解 矩阵的绝对值 矩阵的谱分解 正规矩阵 

分 类 号：O151.21[理学—数学]