In this paper, we establish the theory of tetragonal curves and present a systematic method for constructing Riemann theta function solutions to algebro-geometric initial value problems for matrix mKdV equations. Starting from a \(4\times 4\) matrix spectral problem, we derive Lax pairs of matrix mKdV equations using the zero-curvature equation and Lenard equations. The corresponding tetragonal curve is introduced via the characteristic polynomial of the Lax matrices of the hierarchy. Then we discuss Riemann theta functions, the construction of the basis of holomorphic differentials, as well as Abelian differentials of the second and third kinds. Based on the theory of tetragonal curves, we analyze algebro-geometric properties of Baker-Akhiezer functions and a class of meromorphic functions. Finally, we obtain Riemann theta function solutions for the whole matrix mKdV hierarchy through asymptotic analysis.
