Extreme Spectra Realization by Nonsymmetric Tridiagonal and Nonsymmetric Arrow Matrices

We consider the following inverse extreme eigenvalue problem: given the real numbers {λ1j,λjj}j=1n and the real vector x(n)=x1,x2,…,xn, to construct a nonsymmetric tridiagonal matrix and a nonsymmetric arrow matrix such that {λ1j,λjj}j=1n are the minimal and the maximal eigenvalues of each one of their leading principal submatrices, and x(n),λn(n) is an eigenpair of the matrix. We give sufficient conditions for the existence of such matrices. Moreover our results generate an algorithmic procedure to compute a unique solution matrix.

Inverse eigenvalue problems for acyclic matrices whose graph is a dense centipede

The reconstruction of a matrix having a pre-defined structure from given spectral data is known as an inverse eigenvalue problem (IEP). In this paper, we consider two IEPs involving the reconstruction of matrices whose graph is a special type of tree called a centipede. We introduce a special type of centipede called dense centipede.We study two IEPs concerning the reconstruction of matrices whose graph is a dense centipede from given partial eigen data. In order to solve these IEPs, a new system of nomenclature of dense centipedes is developed and a new scheme is adopted for labelling the vertices of a dense centipede as per this nomenclature . Using this scheme of labelling, any matrix of a dense centipede can be represented in a special form which we define as a connected arrow matrix. For such a matrix, we derive the recurrence relations among the characteristic polynomials of the leading principal submatrices and use them to solve the above problems. Some numerical results are also provided to illustrate the applicability of the solutions obtained in the paper.

Extremal Inverse Eigenvalue Problem for a Special Kind of Matrices

We consider the following inverse eigenvalue problem: to construct a special kind of matrix (real symmetric doubly arrow matrix) from the minimal and maximal eigenvalues of all its leading principal submatrices. The necessary and sufficient condition for the solvability of the problem is derived. Our results are constructive and they generate algorithmic procedures to construct such matrices.