We introduce matrix $Φ$-ribbons, extending the scalar $Φ$-ribbons to the matrix case. For $Φ$ in the subadditivity class, we establish data processing and tensorization properties of matrix $Φ$-ribbons. We establish scalar and matrix $Φ$-entropy inequalities and use them to show that both $Φ$-ribbons characterize the corresponding $Φ$-strong data processing inequality (SDPI) constants. We obtain exact evaluations of the SDPI constants for several sources. We provide examples showing that the matrix $Φ$-SDPI constant can exceed the scalar counterpart. Finally, we derive inner bounds for $Φ$-ribbons from prescribed independence structures and prove that these bounds are optimal when $Φ(t)=t\log t$.

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