  We define geometric matrix midranges for positive definite Hermitian matrices
and study the midrange problem from a number of perspectives. Special attention
is given to the midrange of two positive definite matrices before considering
the extension of the problem to $N > 2$ matrices. We compare matrix midrange
statistics with the scalar and vector midrange problem and note the special
significance of the matrix problem from a computational standpoint. We also
study various aspects of geometric matrix midrange statistics from the
viewpoint of linear algebra, differential geometry and convex optimization.
