The inverse problem of a generalized Jacobi matrix
AI Shu-li · Basic Sciences Journal of Textile Universities · 2008
The inverse problem of weighted Jacobi matrices was considered.Based on gradually discussion,in terms of eigenpairs,a necessary and sufficient condition is given so that the solution of the generalized inverse problem of a Jacobi matrix is determined uniquely and the Jacobi matrix is constructed from its generalized eigenpairs.That is,when i=1,2,…,k-1,if Di≠0,[(μ1-λ)di+λqiDi+(μ1-λ)Mi-1+(μ1-λ)qixiyi+1]/Di0,then bi=[(μ1-λ)di+λqiDi+(μ1-λ)Mi-1+(μ1-λ)qixiyi+1]/Di,ai=λpi+[λqi-1-bi-1)xi-1+(λqi-bi)xi+1]/xi,xi≠0,μipi+[(μ1qi-1-bi-1)yi-1+(μ1qi-bi)yi+1]/yi,xi=0.If Di=0,then,bi=(μ1qi-1yi-1+μ1qiyi+1-bi-1yi-1)/yi+1,and ai can be any real number.For i=k,k+1,…,n-1,ai,bi can be calculated similarly.