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Let $s\in{\mathbb R}$, $q\in (0,\infty]$, and $\tau\in[0,\infty)$. It is wellknown that Besov-type spaces $\dot B^{s,\tau}_{p,q}$ with $p\in (0,\infty]$ andTriebel--Lizorkin-type spaces $\dot F^{s,\tau}_{p,q}$ with $p\in (0,\infty)$when $\tau\in [0,\infty)$ or with $p\in (0,\infty]$ when $\tau=0$ on$\mathbb{R}^n$ consist of a general family of function spaces that cover notonly the well-known Besov and Triebel--Lizorkin spaces $\dot B^{s}_{p,q}$ and$\dot F^{s}_{p,q}$ (when $\tau=0$) but also several other function spaces ofinterest, such as Morrey spaces and $Q$ spaces. In three successive articles,the authors develop a complete real-variable theory of matrix-weightedBesov-type spaces $\dot B^{s,\tau}_{p,q}(W)$ and matrix-weightedTriebel--Lizorkin-type spaces $\dot F^{s,\tau}_{p,q}(W)$ on $\mathbb{R}^n$,where $W$ is a matrix-valued Muckenhoupt $A_p$ weight. This article is thefirst one, whose main novelty exists in that the authors introduce the newconcept, $A_p$-dimensions of matrix weights, and intensively study theirproperties, especially those elaborate properties expressed via reducingoperators. The authors then introduce the spaces $\dot B^{s,\tau}_{p,q}(W)$ and$\dot F^{s,\tau}_{p,q}(W)$ and, using $A_p$-dimensions and their niceproperties, the authors establish the $\varphi$-transform characterization of$\dot B^{s,\tau}_{p,q}(W)$ and $\dot F^{s,\tau}_{p,q}(W)$. The $A_p$-dimensionsof matrix weights and their properties also enable the authors to obtain thesharp boundedness of almost diagonal operators on related sequence spaces inthe subsequent second article and the optimal characterizations of moleculesand wavelets, trace theorems, and the optimal boundedness ofpseudo-differential operators and Calder\'on--Zygmund operators in thesubsequent third article.

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces I: $A_p$-Dimensions of Matrix Weights and $\varphi$-Transform Characterizations