Matrix Padé-type approximant and directional matrix Padé approximant in the inner product space

A new matrix Padé-type approximant (MPTA) is defined in the paper by introducing a generalized linear functional in the inner product space. The expressions of MPTA are provided with the generating function form and the determinant form. Moreover, a directional matrix Padé approximant is also established by giving a set of linearly independent matrices. In the end, it is shown that the method of MPTA can be applied to the reduction problems of the high degree multivariable linear system.