We characterize the class $CG(s)$ of matrices A for which the linear system A x = b can be solved by an s-term conjugate gradient method. We show that, except for a few anomalies, the class $CG(s)$ consists of matrices A for which conjugate gradient methods are already known. These matrices are the Hermitian matrices, A^ * = A, and the matrices of the form Aeiθ (dI + B), with B^ * = - B.
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Faber et al. (1984) studied this question.