  Given a matrix-weight $W$ in the Muckenhoupt class
$\mathbf{A}_p(\mathbb{R}^n)$, $1\leq p<\infty$, we introduce corresponding
vector-valued continuous and discrete $\alpha$-modulation spaces
$M^{s,\alpha}_{p,q}(W)$ and $m^{s,\alpha}_{p,q}(W)$ and prove their equivalence
through the use of adapted tight frames. Compatible notions of molecules and
almost diagonal matrices are also introduced, and an application to the study
of pseudo-differential operators on vector valued spaces is given.
