Sorensen's iteratively restarted Arnoldi algorithm is one of the most successful and flexible methods for finding a few eigenpairs of a large matrix. However, the need to preserve structure of the Arnoldi decomposition, on which the algorithm is based, restricts the range of transformations that can be performed on it. In consequence, it is difficult to deflate converged Ritz vectors from the decomposition. Moreover, the potential forward instability of the implicit QR algorithm can cause unwanted Ritz vectors to persist in the computation. In this paper we introduce a generalized Arnoldi decomposition that solves both problems in a natural and efficient manner. (Also cross-referenced as UMIACS-TR-2000-21)