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We consider the Gaussian multiple-input multiple-output (MIMO) broadcast channel with common and private messages. We obtain an outer bound for the capacity region of this channel. To this end, we show that a parallel Gaussian broadcast channel can be constructed from any given Gaussian MIMO broadcast channel by using the generalized singular value decomposition and a relaxation on the power constraint for the channel input. Due to this relaxation of the power constraint, the capacity region of the constructed parallel channel, which is known, provides an outer bound for the capacity region of the original channel. We show that this outer bound is within a finite gap of the capacity region by comparing it with an achievable rate region that can be obtained either by using dirty-paper coding or by using a variation of the zero-forcing scheme.