U [??] [PHI], is called the linear equivalence transformation.

The application of the linear equivalence transformation U to equations (2) yields the system in the strong Popov form (3) with respect to the inputs.

Due to Assumption 2 one can transform (9) via linear equivalence transformation to

It is clear that the transformation introduced in Theorem 1.1 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 (2.4)), and the non-structured equivalence transformation (with (2.7)).

* [Eu.sub.max], [Es.sub.max], [Eh.sub.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.

It is shown that the singular LQ problem for irregular singular systems with persistent disturbances can be transformed to the optimal problem for standard state space systems by restricted system equivalence transformation. The system state is decomposed into free state and restricted state and the input is decomposed into free input and forced input.

By restricted system equivalence transformation, we transformed the singular LQ problem for irregular singular systems with persistent disturbances to the optimal problem for standard state space systems.

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.

The transformations are the equivalence transformations that, by definition, do not change the zeros (solutions) of the system equations.

Of course, additional denominators that show up in the reduction algorithm should also be included in S together with their shifts and powers; this means denominators both in the equivalence transformations and in transformed matrix W.