Frederik van Schagen

Full Width Blocks

This chapter presents the analogues of the results of the two previous chapters for full width blocks. The first three sections extend the results of Chapter III. The fourth section explains the duality between full length and full width blocks, and provides the tools to transfer the results of Chapter IV to full width blocks, which is done in the last two sections.

Connections with Linear Systems

This chapter gives an introduction to the theory of discrete time-invariant linear input/output systems. Included is the analysis of observability and controllability of systems, theorems about minimal realization, the eigenvalue assignment theorem and state feedback, and theorems on output stabilization and output injection. The connection with operator blocks and block similarity theory is one of the main features of this chapter. For example, observability indices and controllability indices are identified as block similarity invariants of a certain kind, state feedback is viewed as block similarity, and the eigenvalue assignment problem is treated as an eigenvalue completion problem. These connections are exploited throughout the chapter.

Applications to Matrix Polynomials

This chapter contains two applications to the theory of regular matrix polynomials. The first is connected with the zero structure, and relates the behaviour of a matrix polynomial at infinity to block indices of a certain kind. The second deals with Wiener-Hopf factorization and identifies the factorization indices as block similarity invariants.

Factorization of Analytic Operator Functions

This chapter carries out a similar program as in the previous chapter, but now for analytic operator-valued functions. Wiener-Hopf equivalence and Wiener-Hopf factorization are described in terms block similarity of certain operator blocks and their canonical forms.

Elementary Operations on Blocks

In this chapter we introduce the notions of block-invariant subspace and of direct sum of blocks. Together with block similarity these notions provide the main tools for analyzing the structure of block decompositions. A list (which will later be shown to be a full list) of indecomposable blocks is given.

The Eigenvalue Completion Problem for Full Length Blocks

In this chapter we give the solution of the eigenvalue completion problem for full length blocks. The proof is based on a reduction to a matrix polynomial completion problem. The related full length restriction problem is also solved. Applications to matrix pencils and non—everywhere defined operators on finite dimensional spaces are included. Another treatment of the problem, in terms of a certain matrix equation, is also presented.

Full Length Blocks

In this chapter we give the decomposition in indecomposable blocks and describe the invariants under block similarity for full length blocks. Applications to matrix pencils and to non—everywhere defined linear operators on finite dimensional spaces are included. In the last section we bring together some results on non—increasing sequences of nonnegative numbers, that are used in several places in this chapter and elsewhere in the book.

Infinite Dimensional Operator Blocks

In this chapter we develop the theory of operator blocks and block similarity on infinite dimensional spaces. We do not treat the general case, but only consider the analogues of full width and full length blocks. As an application a solution of the Hilbert space analogue of the eigenvalue completion problem is obtained for full length blocks.

Applications to Rational Matrix Functions

In this chapter we continue with applications of the general theory of operator blocks. We deal with two kind of homogeneous interpolation problems and with factorization indices for regular rational matrix functions. The first section contains preliminary material on the zero and pole structure of rational matrix functions.

Eigenvalue Completion Problems for Triangular Matrices

In this chapter the theory developed in the previous chapters is extended to a class of partially specified matrices with a non-block pattern. It deals with matrices of which the upper triangular part is given and the elements in the strictly lower triangular part are considered as unspecified. For such partially specified matrices we consider problems that are similar in nature to the ones considered earlier for operator blocks. The class of admissible similarities consists here of the U-similarities, which are the natural analogies of block similarities in the context of this chapter. In principle, we carry out the same program as for operator blocks. For example, we describe the invariants and the canonical form for certain equivalence classes under U-similarity. The analogue of the eigenvalue completion problem appears in a natural way and is solved for the case when the multiplicities are not taken into account. As an application a full solution of the spectral radius completion problem is obtained in this triangular setting.