On some structured inverse eigenvalue problems

TL;DR: The connection between the two algorithms exhibits a similarity transformation from the classical Frobenius companion matrix to the tridiagonal matrix, used to illustrate the fact that, when computing the eigenvalues of a matrix, the nonsymmetric Lanczos algorithm may lead to a slow convergence, even for a symmetric matrix.

Abstract: This work deals with various finite algorithms that solve two special Structured Inverse Eigenvalue Problems (SIEP). The first problem we consider is the Jacobi Inverse Eigenvalue Problem (JIEP): given some constraints on two sets of reals, find a Jacobi matrix J (real, symmetric, tridiagonal, with positive off-diagonal entries) that admits as spectrum and principal subspectrum the two given sets. Two classes of finite algorithms are considered. The polynomial algorithm which is based on a special Euclid–Sturm algorithm (Householder's terminology) and has been rediscovered several times. The matrix algorithm which is a symmetric Lanczos algorithm with a special initial vector. Some characterization of the matrix ensures the equivalence of the two algorithms in exact arithmetic. The results of the symmetric situation are extended to the nonsymmetric case. This is the second SIEP to be considered: the Tridiagonal Inverse Eigenvalue Problem (TIEP). Possible breakdowns may occur in the polynomial algorithm as it may happen with the nonsymmetric Lanczos algorithm. The connection between the two algorithms exhibits a similarity transformation from the classical Frobenius companion matrix to the tridiagonal matrix. This result is used to illustrate the fact that, when computing the eigenvalues of a matrix, the nonsymmetric Lanczos algorithm may lead to a slow convergence, even for a symmetric matrix, since an outer eigenvalue of the tridiagonal matrix of order n − 1 can be arbitrarily far from the spectrum of the original matrix.