The Poincaré metric on the unit disk is invariant under conformal maps of the disk onto itself and contracted by any analytic map of the disk into itself. On a domain G in several complex dimensions there are many invariant metrics and some of these are contracted and some are not. We are led by a natural question in engineering to study \(G={\mathcal B}M_ n\)- the unit ball of \(n\times n\) matrices and the biholomorphic invariants produced by the 'cross ratio' of two matrices (this includes all of the classical metrics which we know on \({\mathcal B}M_ n)\). 1. The paper characterizes all analytic maps of \({\mathcal B}M_ n\) into itself which contract the invariants produced by the cross ratio. In particular we prove that the linear fractional maps \({\mathcal F}(S)=(AS+B)(CS+D)^{-1}\) with matrix coefficients A,B,C,D which map \({\mathcal B}M_ n\) into itself have this property. For example, they contract the Siegel metric. 2. We actually do a more general study involving the cross ratio of a matrix in \({\mathcal B}M_ n\) with an arbitrary matrix and prove that linear fractional maps contract the invariants in the more general situation. Suppose one is given an amplifying device A and wants to use it as the central component in an amplifier. The objective is to design passive circuitry (or lossless circuitry) which when connected with A results in an amplifier with large gain. A natural question is: can one build a better amplifier (more gain) with passive connecting circuitry than with lossless connecting circuitry? This paper proves that the answer is no. In fact, this physical statement is the precise content of 1. and 2. - 1. proves it for a reflection type amplifier and 2. proves it for a (linearized) transistor amplifier.