Eigenvalue Completion Problems for Triangular Matrices

Abstract

In this chapter the theory developed in the previous chapters is extended to a class of partially specified matrices with a non-block pattern. It deals with matrices of which the upper triangular part is given and the elements in the strictly lower triangular part are considered as unspecified. For such partially specified matrices we consider problems that are similar in nature to the ones considered earlier for operator blocks. The class of admissible similarities consists here of the U-similarities, which are the natural analogies of block similarities in the context of this chapter. In principle, we carry out the same program as for operator blocks. For example, we describe the invariants and the canonical form for certain equivalence classes under U-similarity. The analogue of the eigenvalue completion problem appears in a natural way and is solved for the case when the multiplicities are not taken into account. As an application a full solution of the spectral radius completion problem is obtained in this triangular setting.