A Note on the Symmetric Recursive Inverse Eigenvalue Problem

In [1] the recursive inverse eigenvalue problem for matrices was introduced. In this paper we examine an open problem on the existence of symmetric positive semidefinite solutions that was posed there. We first give several counterexamples for the general case and then characterize under which further assumptions the conjecture is valid. 1. Introduction. In [1] several classes of recursive inverse eigenvalue problems were introduced that construct matrices from eigenvalues and eigenvectors of leading principal submatrices. A simple application of such problems is the construction of Leontief models in economics, see e.g., [2], when a feasible model with n − 1 inputs and n − 1 outputs is extended (by adding an input and an output) to a larger feasible model with prescribed equilibrium point, see [1]. In this paper we discuss the particular case of the real symmetric recursive inverse eigenvalue problem, in the following denoted by SRIEP(n) which has the following form: