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This chapter links Krylov subspace methods to classical topics of linear algebra. The main goal is to explain when a Krylov sequence can be orthogonalised with an optimal (Arnoldi-type) short recurrence. This question was posed by Golub in 1981 and answered by the Faber–Manteuffel theorem in 1984. The chapter gives a new complete proof of this theorem that has not been published elsewhere. It is based on the cyclic decomposition of a vector space with respect to a given linear operator. The theorem motivates the theoretically and practically important distinction made between Hermitian and non-Hermitian problems in the area of Krylov subspace methods. The matrix-version of the theorem works with the so-called B-normal(s) matrices, and this property is linked for a general matrix with the number of its distinct eigenvalues. The chapter also reviews other types of recurrences and it ends with brief remarks on integral representations of invariant subspaces.

*Keywords: *
cyclic invariant subspaces;
Jordan canonical form;
length of Krylov sequences;
optimal short recurrences;
Faber–Manteuffel theorem;
b-normal(s) property;
harmonic polynomials;
isometric Arnoldi algorithm;
generalised Lanczos algorithm;
cauchy integral representation

*Chapter.*
*35102 words.*
*Illustrated.*

*Subjects: *
Applied Mathematics

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