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We propose a scalar upper bound on the capacity region of the isotropic fading vector broadcast channel in terms of the capacity region of a scalar fading broadcast channel. The scalar upper bound is applicable to the broad class of isotropic fading broadcast channels regardless of the distribution of the users' channel magnitudes, the distribution of the additive noise experienced by each user, or the amount of channel knowledge available at the receiver. Using this upper bound, we prove the optimality of the Alamouti scheme in a broadcast setting, extend the recent results on the capacity of nondegraded, fading scalar broadcast channels to nondegraded fading vector broadcast channels, and determine the capacity region of a fading vector Gaussian broadcast channel with channel magnitude feedback. We also provide an example of a Rayleigh-fading broadcast channel with no channel state information available to the receiver (CSIR), where the bound on the capacity region obtained by a naive application of the scalar upper bound is provably loose, because it fails to account for the additional loss in degrees of freedom due to lack of channel knowledge at the receiver. A tighter upper bound is obtained by separately accounting for the loss in degrees of freedom due to lack of CSIR before applying the scalar upper bound.