An i/o equivalence transformation for system (1) is an invertible transformation of the system equations to another set of equations of the form (1), being i/o equivalent with the original system equations.

Then the equivalence transformation of system (1) can be found by solving the system of partial differential equations (17), resulting in the new system having the same row orders, except the (i + 1)th one which, by (18), is strictly less than [beta]i+1.

Under Assumption 3 there exists a (local) equivalence transformation that allows one to transform the set of equations (1) into a strong row-reduced form, possibly together with some equations which are trivially satisfied, or define restrictions on input or output signals (20).

The first example shows that the approach proposed in this paper in some cases yields the same linear i/o equivalence transformation as the method from [5].

A particular solution, based on linear i/o equivalence transformations, was proposed in [5].

Based on the reasons mentioned above, the main goals of this paper are: to present a new definition of the strong row-reducedness property of a system and to specify a larger class of local nonlinear i/o equivalence transformations.

1 preserves the structure of the pencil, but note that in the real case or in the complex case with * being the complex conjugate, this transformation is a congruence transformation, while in the complex case with * being the transpose, this is just a structure preserving equivalence transformation but not a congruence transformation.

In this section we present several numerical tests to compare the computed finite eigenvalues of the subpencils generated by the three methods: structured unitary equivalence transformation (Algorithms 1 and 2), Schur complement transformation (by using (2.

max]: the maximum relative error of the finite eigenvalues for a given pencil with the structured unitary equivalence transformation method, the Schur complement method, and the non-structured equivalence transformation method, respectively.

min]: the minimum relative error of the finite eigenvalues for a given pencil with the structured unitary equivalence transformation method, the Schur complement method, and the non-structured equivalence transformation method, respectively.

GM]: the geometric mean of the relative errors of the finite eigenvalues for a given pencil with the structured unitary equivalence transformation method, the Schur complement method, and the non-structured equivalence transformation method, respectively.

hat]: the maximum real part of the eigenvalues computed by the non-structured equivalence transformation method in the case when all the exact finite eigenvalues are purely imaginary.