Four-dimensional Painlev\'e-type equations associated with ramified linear equations I: Matrix Painlev\'e systems

As a sequel to Kawakami-Nakamura-Sakai (arXiv:1209.3836), this series of papers constructs the complete degeneration scheme of four-dimensional Painlev'e-type equations which includes the Painlev'e-type equations associated with linear systems of ramified type. In the present paper, we consider the degeneration of Painlev'e-type equations which we call the matrix Painlev'e systems.