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

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