This paper deals with the block partial fraction expansion of a matrix transfer function F(s), where F(s) = Nr(s)Dr-1(s) = Dl-1(s) and Nr(s), Dr(s), Dl (s) and Nl (s) are ??-matrices with matrix coefficients. A new algorithm is derived to construct a transformation matrix that transforms a right (left) solvent to the corresponding left (right) solvent of a matrix polynomial. Also, the algorithm can be used to construct a set of right (left) fundamental matrix polynomials and the inversion of a block Vandermonde matrix. A new technique is derived to perform the block partial fraction expansion of a matrix transfer function. The developed algebraic theory enhances the capability of the analysis and synthesis of a class of multivariable systems described by high degree matrix differential equations.