The Problem Of A Class Of Constrained Matrix Equation And The Problem Of A Class Of Matrix Extension

The problem of constrained matrix equation is to find solutions of matrix equations in a set of matrices satisfying some constrained conditions. The problem of matrix extension is to construct matrices with a given matrix or a several given matrices as its sub-matrices under some constrained conditions. The problem of constrained matrix equation and the problem of matrix extension come from many fields, which have been widely used in structural design, parameter identification, biology, electricity, molecular spectroscopy, solid mechanics, automatic control theory, vibration theory, finite elements, linear optimal control and so on. This thesis mainly discusses the following problems.ProblemⅠGiven M∈Rn×k, N∈Rl×n, A∈Rm×k, B∈Rl×p, C∈Rm×p, X~∈Rk×land L ? Rk×l, let L1 = {X :X∈L,AXB?C=min}. Find X?∈L1, such that where ? is Frobenius norm.ProblemⅡGiven A∈Rm×n, B∈Rk×l, C∈Rm×l, X 0∈Rp×q(1≤p≤n,1≤q≤k), X~∈Rn×kand S ?Rn×k, let Find X?∈S1, such that where X ([1 :p,1:q]) is on the above left p×q sub-matrix of matrix X .This thesis will discuss ProblemⅠaboutⅠ~Ⅲgeneralized symmetric(generalized anti-symmetric) matrix.Ⅰ-generalized symmetric(generalized anti-symmetric) matrix means given M∈Rn×k, real n×n matrix A can satisfy with Ax, y= x,Ay( Ax, y = ?x,Ay) for any x∈Rn and y∈R(M);Ⅱ-generalized symmetric (generalized anti-symmetric) matrix means given M∈Rn×k, real n×n matrix A can satisfy with Ax, y= x,Ay( Ax, y= ?x,Ay) for any x∈R(M) and y∈R(M);...