项目名称: 谱范数下矩阵的广义最小秩逼近问题及应用
项目编号: No.11301247
项目类型: 青年科学基金项目
立项/批准年度: 2014
项目学科: 数理科学和化学
项目作者: 李莹
作者单位: 聊城大学
项目金额: 22万元
中文摘要: 本项目拟对谱范数下矩阵的多种类型的广义最小秩逼近问题进行研究,即确定谱范数下广义矩阵逼近问题的最小秩解。主要研究以下五种条件下最小秩广义逼近问题解的存在条件,在可行集非空的条件下,计算约束条件A-BXC的谱范数的最小值,优化目标矩阵X的最小秩及达到秩最小时X的表达式,具体为:①当约束条件‖A-BXC‖为对称形式时,即A为Hermitian矩阵,C为B的共轭转置,要求矩阵X满足半正定条件;②目标优化矩阵X满足正规条件;③矩阵X满足子矩阵约束条件;④将约束条件中的单变量函数推广为多变量函数;⑤将约束条件中的包含优化目标矩阵X的矩阵函数作为某一矩阵M的一个子块,要求M的谱范数达到极小。另外,针对①中的问题,设计高效稳定的算法,寻求其在控制中的应用。该项目的研究可为从复杂数据中寻找最有用的信息提供可靠的应用基础。
中文关键词: 矩阵逼近;四元数矩阵;保结构算法;实表示矩阵;矩阵方程
英文摘要: This project studies the problem of minimum rank generalized approximation of matrix in spectral norm.That is to say,we will characterize the expressions of the minimum value of spectral norm in constraint condition,the minimum rank of X and derive a general form of minimum rank solution X. It mainly includes the following contents. First, constraint condition is Hermitian and objective optimization matrix is semi-positive definite.Second, objective optimization matrix X is normal.Third,X is a matrix with submatrix constraint . Forth, the problem of minimum rank generalized approximation of matrix in spectral norm under multivariable constraint conditions.Fifth, a matrix function of objective optimization matrix is a subblock of a matrix M, and the spectral norm of M is minimum or less than a given value. For the first question ,we will give the rapid and stable algorithms and apply them to solve practical problem in the control theory.
英文关键词: matrix approxiamation;quaternion matrix;structure-preserving algorithm;real representation matrix;matrix equation