  We consider the problem of approximation of matrix functions of class $L^p$
on the unit circle by matrix functions analytic in the unit disk in the norm of
$L^p$, $2\le p<\be$. For an $m\times n$ matrix function $\Phi$ in $L^p$, we
consider the Hankel operator $H_\Phi:H^q(C^n)\to H^2_-(C^m)$, $1/p+1/q=1/2$. It
turns out that the space of $m\times n$ matrix functions in $L^p$ splits into
two subclasses: the set of respectable matrix functions and the set of weird
matrix functions. If $\Phi$ is respectable, then its distance to the set of
analytic matrix functions is equal to the norm of $H_\Phi$. For weird matrix
functions, to obtain the distance formula, we consider Hankel operators defined
on spaces of matrix functions. We also describe the set of $p$-badly
approximable matrix functions in terms of special factorizations and give a
parametrization formula for all best analytic approximants in the norm of
$L^p$. Finally, we introduce the notion of $p$-superoptimal approximation and
prove the uniqueness of a $p$-superoptimal approximant for rational matrix
functions.
