6.2 Krylov Subspaces

In this section we consider projection methods on Krylov subspaces, i.e., subspaces of the form

which will be denoted simply by if there is no ambiguity. The dimension of the subspace of approximants increases by one at each step of the approximation process. A few elementary properties of Krylov subspaces can be established. A first property is that is the subspace of all vectors in that can be written as x = p(A) ?, where p is a polynomial of degree not exceeding m ? 1. Recall that the minimal polynomial of a vector v is the nonzero monic polynomial p of lowest degree such that p(A) ? = 0. The degree of the minimal polynomial of ? with respect to A is often called the grade of ? with respect to A or simply the grade of ? if there is no ambiguity. A consequence of the Cayley-Hamilton theorem is that the grade of ? does not exceed n. The following proposition is easy to prove.