Let
$\Phi $ be an algebraically closed field of characteristic zero,
$G$ a finite, not necessarily abelian, group. Given a
$G$-grading on the full matrix algebra
$A\,=\,{{M}_{n}}\left( \Phi \right)$, we decompose
$A$ as the tensor product of graded subalgebras
$A\,=\,B\,\otimes \,C,\,B\,\cong \,{{M}_{p}}\left( \Phi \right)$ being a graded division algebra, while the grading of
$C\,\cong \,{{M}_{q}}\left( \Phi \right)$ is determined by that of the vector space
${{\Phi }^{n}}$. Now the grading of
$A$ is recovered from those of
$A$ and
$B$ using a canonical “induction” procedure.