Preliminaries

In this section three important concepts will be discussed, essential for the remainder of the manuscript. We will discuss in the following order: a formal definition of Hessenberg-like matrices, an efficient representation for these matrices, interactions between Givens transformations, the $QR$-factorization of Hessenberg-like matrices and briefly the multishift setting.

It might seem at first that the introduced concepts are somewhat independent of each other, but pieces will fall together in the upcoming sections.