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This is the last one of three successive articles by the authors onmatrix-weighted Besov-type and Triebel--Lizorkin-type spaces $\dotB^{s,\tau}_{p,q}(W)$ and $\dot F^{s,\tau}_{p,q}(W)$. In this article, theauthors establish the molecular and the wavelet characterizations of thesespaces. Furthermore, as applications, the authors obtain the optimalboundedness of trace operators, pseudo-differential operators, andCalder\'on--Zygmund operators on these spaces. Due to the sharp boundedness ofalmost diagonal operators on their related sequence spaces obtained in thesecond article of this series, all results presented in this article improvetheir counterparts on matrix-weighted Besov and Triebel--Lizorkin spaces $\dotB^{s}_{p,q}(W)$ and $\dot F^{s}_{p,q}(W)$. In particular, even when revertingto the boundedness of Calder\'on--Zygmund operator on unweightedTriebel--Lizorkin spaces $\dot F^{s}_{p,q}$, these results are still better.

Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces III:  Characterizations of Molecules and Wavelets, Trace Theorems, and Boundedness  of Pseudo-Differential Operators and Calder\'on--Zygmund Operators