  Let $\alpha\in\mathbb R$, $q\in(0,\infty]$, $p\in(0,\infty)$, and $W$ be an
$A_p(\mathbb{R}^n,\mathbb{C}^m)$-matrix weight. In this article, the authors
characterize the matrix-weighted Triebel-Lizorkin space
$\dot{F}_{p}^{\alpha,q}(W)$ via the Peetre maximal function, the Lusin area
function, and the Littlewood-Paley $g_{\lambda}^{*}$-function. As applications,
the authors establish the boundedness of Fourier multipliers on matrix-weighted
Triebel-Lizorkin spaces under the generalized H\"ormander condition. The main
novelty of these results exists in that their proofs need to fully use both the
doubling property of matrix weights and the reducing operator associated to
matrix weights, which are essentially different from those proofs of the
corresponding cases of classical Triebel-Lizorkin spaces that strongly depend
on the Fefferman-Stein vector-valued maximal inequality on Lebesgue spaces.
