Factorization of non-proper rational matrix functions

Abstract

In this chapter we treat the problem of factorizing a non-proper rational matrix function. The realization used in the earlier chapters is replaced by

$$ V(\lambda ) = I + C(\lambda G - A)^{ - 1} B. $$

Here I=I _m is the m × m identity matrix, A and G are square matrices of order n say, and the matrices C and B are of sizes m × n and n × m, respectively. Any rational m × m matrix function W, proper or non-proper, admits such a representation. The representation (4.1) allows us to extend the results obtained in the previous chapter to arbitrary rational matrix functions. As an application we treat the problem to invert a singular integral operator with a rational matrix symbol.