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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.

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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