A generalized-Jordan-form-approach one-step irreducible realization of matrices

This note presents a one-step irreducible realization (
A,B,C,D
) from a rational matrix
G(s)
with multiple poles. The introduction of a generalized-Jordan-form
A
makes it possible to consider additional network synthesis constraints in determining
C
column by column and
B
row by row. Only Gauss-Jordan reduction is needed. One example is given for illustration.