Multiple Recurrences and the Associated Matrix Structures Stemming From Normal Matrices

There are many classical results in which orthogonal vectors stemming from Krylov subspaces are linked to short recurrence relations, e.g., three-terms recurrences for Hermitian and short rational recurrences for unitary matrices. These recurrence coefficients can be captured in a Hessenberg matrix, whose structure reflects the relation between the spectrum of the original matrix and the recurrences. The easier the recurrences, the faster the orthogonal vectors can be computed possibly resulting in computational savings in the design of, e.g., iterative solvers. In this article we focus on multiple recursions, i.e., the (j + 1)st orthogonal vector satisfies