  The orbit space for a scalar field in a complex square matrix representation
obtains a Minkowski space structure from the Cauchy-Schwarz inequality. It can
be used to find vacuum stability conditions and minima of the scalar potential.
The method is suitable for fields such as a bidoublet, an $SU(2)$ triplet or
$SU(3)$ octet. We use the formalism to find the vacuum stability conditions for
the left-right symmetric potential of a bidoublet and left and right Higgs
doublets.
