A new algorithm on the inverse eigenvalue problem for double dimensional Jacobi matrices (Q445837)

Some properties of eigenvalues and eigenvectors of Jacobi matrices are studied. A new algorithm for reconstructing a \(2n\)-th order Jacobi matrix \(J_{2n}\) with a given \(n\)th order leading principal submatrix \(J_{n}\) and with all eigenvalues of \(J_{2n}\) is proposed. This algorithm needs to compute the eigenvalues of the \(n\)th order tailing principal submatrix \(J_{n+1,2n}\) and the first components of the unit eigenvectors of \(J_{n+1,2n}\). The method avoids computing the coefficients of the characteristic polynomial for getting the eigenvalues of \(J_{n+1,2n}\). Some numerical results are presented.