  This paper considers an entropy-power inequality (EPI) of Costa and presents
a natural vector generalization with a real positive semidefinite matrix
parameter. This new inequality is proved using a perturbation approach via a
fundamental relationship between the derivative of mutual information and the
minimum mean-square error (MMSE) estimate in linear vector Gaussian channels.
As an application, a new extremal entropy inequality is derived from the
generalized Costa EPI and then used to establish the secrecy capacity regions
of the degraded vector Gaussian broadcast channel with layered confidential
messages.
