The main aim of this paper is introduced to an extension of the Hermite-Hermite matrix polynomials. An explicit representation, some important addition and multiplication properties, a matrix recurrence relation, in particular the three terms recurrence relation for the an extension of the Hermite-Hermite matrix polynomials are given, matrix partial differential equations satisfied them is presented,
relationships with Hermite-Hermite and Legendre matrix polynomials of two variable, and some special properties. A new expansions of the matrix functions for a wide class of matrices in terms of generalized Hermite-Hermite matrix polynomials is proposed.