  For a Banach space $V$ we define its Lipschitz extension constant,
$\cL\cE(V)$, to be the infimum of the constants $c$ such that for every metric
space $(Z,\rho)$, every $X \subset Z$, and every $f: X \to V$, there is an
extension, $g$, of $f$ to $Z$ such that $L(g) \le cL(f)$, where $L$ denotes the
Lipschitz constant. The basic theorem is that when $V$ is finite-dimensional we
have $\cL\cE(V) = \cP\cC(V)$ where $\cP\cC(V)$ is the well-known projection
constant of $V$. We obtain some direct consequences of this theorem, especially
when $V = M_n(\bC)$. We then apply techniques for calculating projection
constants, involving averaging projections, to calculate
$\cL\cE((M_n(\bC))^{sa})$. We also discuss what happens if we also require that
$\|g\|_{\infty} = \|f\|_{\infty}$.
