  We prove an Asymptotic Implicit Function Theorem in the setting of Gevrey
asymptotics with respect to a parameter. The unique implicitly defined solution
admits a Gevrey asymptotic expansion and furthermore it is the Borel
resummation of the corresponding implicitly defined formal power series
solution. The main theorem can therefore be rephrased as an Implicit Function
Theorem for Borel summable power series. As an application, we give a diagonal
or Jordan decomposition for holomorphic matrices in Gevrey asymptotic families.
