A Two-random Matrix Determined By The Spectrum And A Class Of Matrix Equation Problems

The inverse eigenvalue problem of non-negative matrix has been the focus of research in numerical algebra,and doubly stochastic matrix is one of the most common matrix in matrix inverse eigenvalue problem,So the study of doubly stochastic matrix itself and the characteristic of the spectrum is the basis of the research on the inverse eigenvalue problem.The matrix equation problem is derived from the inverse problem of vibration theory,the main research content is to find a matrix equation of different forms of solution and theirs optimal approximation.And this problem in the mechanical system and civil engineering structure has some practical background.In this paper,we mainly study the inverse eigenvalue problem for a class of special doubly stochastic matrices and a class of matrix equation problems.This paper is divided into three chapters.The first chapter introduces the research significance of this topic,as well as the current research status of the subject.The second chapter researches the inverse eigenvalue problem of doubly stochastic matrix determined by their spectra.This chapter describes the characteristics of the doubly stochastic matrix determined by their spectra starting from the convex polyhedron,and proposes a conjecture for the general characteristics of the n order matrix,finally based on the analysis of characteristics of permutation matrices and their relationship with the two n order doubly stochastic matrices,it is proved that the two kinds of n order doubly stochastic matrices are determined by their spectrum.The third chapter studies the symmetric M symmetric least square solution's of A~TXA=C and its optimal approximation.This chapter obtains the symmetric M symmetric least square solution's of A~TXA=Cby using canonical correlation decomposition in symmetric M symmetric matrices set.Based on this,by using the projection theorem and the generalized singular v-alue decomposition we get the symmetric M symmetric optimal approximation solution according to a given symmetric matrix X~*.The fourth chapter points out the innovation of this paper and gives the prospect of future research.