  We define the vector-valued, matrix-weighted function spaces $\dot{F}^{\alpha
q}_p(W)$ (homogeneous) and $F^{\alpha q}_p(W)$ (inhomogeneous) on
$\mathbb{R}^n$, for $\alpha \in \mathbb{R}$, $0<p<\infty$, $0<q \leq \infty$,
with the matrix weight $W$ belonging to the $A_p$ class. For $1<p<\infty$, we
show that $L^p(W) = \dot{F}^{0 2}_p(W)$, and, for $k \in \mathbb{N}$, that
$F^{k 2}_p(W)$ coincides with the matrix-weighted Sobolev space $L^p_k(W)$,
thereby obtaining Littlewood-Paley characterizations of $L^p(W)$ and $L^p_k
(W)$. We show that a vector-valued function belongs to $\dot{F}^{\alpha
q}_p(W)$ if and only if its wavelet or $\varphi$-transform coefficients belong
to an associated sequence space $\dot{f}^{\alpha q}_p(W)$. We also characterize
these spaces in terms of reducing operators associated to $W$.
