  We determine the rate region of the Gaussian scalar-help-vector source-coding
problem under a covariance matrix distortion constraint. The rate region is
achieved by a Gaussian achievable scheme. We introduce a novel outer bounding
technique to establish the converse of the main result. Our approach is based
on lower bounding the problem with a potentially reduced dimensional problem by
projecting the main source and imposing the distortion constraint in certain
directions determined by the optimal Gaussian scheme. We also prove several
properties that the optimal solution to the point-to-point rate-distortion
problem for a vector Gaussian source under a covariance matrix distortion
constraint satisfies. These properties play an important role in our converse
proof. We further establish an outer bound to the rate region of the more
general problem in which there are distortion constraints on both the sources.
The outer bound is partially tight in general. We also study its tightness in
some nontrivial cases.
