Inverse Problem Of Eigenvalue And Optimal Approximation For Real Persymmetric Band Matrix

The eigenvalue inverse problem of matrix is to construct a matrix from the partial or complete information of eigenvalues or eigenvectors.The eigenvalue inverse problem of matrix and its optimal approximation on the basis of spectral restriction have been Widely used in structure analysis, molecular spectroscopy,geophysics, electrics,optics,automatic control theory and so on.This paper studies the eigenvalue inverse problem and optimal approximation for real persymmetric band matrix systematically.The following several inverse problems of matrices are mainly discussed:Problem Ⅰ Given r+1real differ numbers λ1,...,λr+1and n-dimension nonzero real vectors x1,...,xr+1,find a real persymmetric band matrix P∈Rn×n which band is2r+1,such that Pxi=λixi(i=1,...,r+1).Problem Ⅱ Given2r+1n-dimension nonzero real vector pairs(y1,z1),.(yr+1,zr+1),find a real persymmetric band matrix P∈Rn×n which band is2r+1,such that P(y1,...,y2r+1)=(z1,...,z2r+1).Problem Ⅲ Given m real differ numbers λ1,...λm and n-dimension nonzero real vectors x1,...,xm,where X=[x1,x2,...,xm],Λ=diag(λ1,λ2,...,λm),find a real persymmetric band matrix P∈Pn×n which band is2r+1,such that PX=XΛ.Problem Ⅳ Given a real persymmetric band matrix P,find a matrix P∈SP such that Where SP is the set of the solutions of Problem Ⅲ,i.e SP={P|PX=XΛ,P∈PSRn×n}.‖.‖F,means Frobenius norm,PSRn×n means the set of all the real persymmetric band matrix P∈Rn×n which band is2r+1.The main results of this paper are as follows: 1. According to the different band of the real persymmetric band matrix, the second and third sections of the second chapter discuss the Problem Ⅰ and Ⅱ in different cases which are r=1,r=n-1,1<r<n-1and r=1,1<r≤n-1. By using the solvable conditions of the linear equation systems, the sufficient and necessary conditions for the solutions of Problem Ⅰ and Ⅱ, the expressions of the solutions and the corresponding numerical algorithm and examples are given in the different cases.2. According to the different band of the real persymmetric band matrix,the second section of the third chapter discuss the Problem Ⅲ and Ⅳ in two cases which are r=n-1,1≤r<n-1.The sufficient and necessary conditions for the solutions of Problem III and IV,the expressions of the solutions and the corresponding numerical algorithm and examples are given respectively in the third section of the third chapter..