Orthogonal Hessenberg reduction and orthogonal Krylov subspace bases. (English) Zbl 1086.65028

Liesen, Jörg; Saylor, Paul E.

SIAM J. Numer. Anal. 42, No. 5, 2148-2158 (2005).

The authors investigate necessary and sufficient conditions for a matrix \(A\) (nonsingular, non-Hermitian) to be \(B\)-reducible to upper Hessenberg form \(H\) with only \(s\) upper diagonals, i.e., such that there exists a matrix \(V\), satisfying \(V^HBV=I\) with \(B\) positive definite Hermitian, and \(AV=VH\). In the case of a symmetric \(A\) and \(B=I\), the matrix \(H\) is tridiagonal (\(s=1\)) corresponding to the 3-term recurrence relation used in the symmetric Lanczos algorithm.
So, this result is clearly related to necessary and sufficient conditions for the existence of a short-term recurrence relation for \(B\)-orthogonalization of a Krylov subspace basis, a problem solved by V. Faber and T. A. Manteuffel [SIAM J. Numer. Anal. 24, No. 1, 170–187 (1987; Zbl 0613.65030)]. The relation with the Faber-Manteuffel theorem is clarified. The conditions rely strongly on the concept of \(B\)-normality which is analysed in detail. The paper uses only elementary linear algebra.

Reviewer: Adhemar Bultheel (Leuven)

MSC:

65F10	Iterative numerical methods for linear systems
15A21	Canonical forms, reductions, classification
15A23	Factorization of matrices
65F25	Orthogonalization in numerical linear algebra

Keywords:

Krylov subspace methods; matrix decomposition; short-term recurrences; normal matrices; B-normality