  A theory of matrix-valued functions from the matricial Smirnov class ${\goth
N}_n^+({\Bbb D})$ is systematically developed. In particular, the maximum
principle of
  V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization
of outer functions and an analogue of Frostman's theorem are presented for
matrix-valued functions from the Smirnov class ${\goth N}_n^+({\Bbb D})$.
  We also consider a family $F_{\lambda} =F-\lambda I$ of functions belonging
to the matricial Smirnov class which is indexed by a complex parameter
$\lambda$. We show that with the exception of a ''very small'' set of such
$\lambda$ the corresponding inner factor in the inner-outer factorization of
the function $F_{\lambda}$ is a Blaschke-Potapov product. The main goal of this
paper is to provide users of analytic matrix-function theory with a standard
source for references related to the matricial Smirnov class.
