# A State Space Approach to Canonical Factorization with by Harm Bart, Israel Gohberg, Marinus A. Kaashoek, André C.M.

By Harm Bart, Israel Gohberg, Marinus A. Kaashoek, André C.M. Ran

The current publication offers with canonical factorization difficulties for di?erent sessions of matrix and operator services. Such difficulties seem in a variety of components of ma- ematics and its functions. The capabilities we examine havein universal that they seem within the country area shape or might be represented in this type of shape. the most effects are all expressed when it comes to the matrices or operators showing within the country house illustration. This comprises priceless and su?cient stipulations for canonical factorizations to exist and particular formulation for the corresponding f- tors. additionally, within the purposes the entries within the nation area illustration play a vital function. Thetheorydevelopedinthebookisbasedonageometricapproachwhichhas its origins in di?erent ?elds. one of many preliminary steps are available in mathematical platforms concept and electric community conception, the place a cascade decomposition of an input-output process or a community is said to a factorization of the linked move functionality. Canonical factorization has an extended and fascinating historical past which begins within the idea of convolution equations. fixing Wiener-Hopf indispensable equations is heavily relating to canonical factorization. the matter of canonical factorization additionally appears to be like in different branches of utilized research and in mathematical structures concept, in H -control concept in particular.

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0 0 . I −q0 Ir −q1 Ir .. −qk−1 Ir ⎤ ⎥ ⎥ ⎥, ⎥ ⎦ ⎡ ⎢ ⎢ B=⎢ ⎢ ⎣ H0 H1 .. Hk−1 ⎤ ⎥ ⎥ ⎥, ⎥ ⎦ C = 0 . . 0 Ir . Then the resolvent set ρ(A) of A coincides with ΩW , the subset of C on which q takes non-zero values. For λ ∈ ρ(A), deﬁne C1 (λ), . . , Ck (λ) by [ C1 (λ) C2 (λ) . . Ck (λ) ] = C(λ − A)−1 . 3. Realization of analytic operator functions 23 From the special form of the matrix A (second companion type) we see that Cj+1 (λ) = λCj (λ), j = 0, . . , k − 1, and C1 (λ) = q(λ)−1 I. Hence C(λ − A) −1 k−1 B = Cj+1 (λ)Hj = j=0 1 H(λ).

6) are matrix-valued realizations. 6), in that order. The counterpart of taking products is factorization. In the next section this topic will be discussed for functions given by a biproper realization. We close the present section with a remark preparing for this discussion. The main operator A in the product realization is given in the form of a 2 × 2 upper triangular operator matrix: A= A1 B1 C2 0 A2 ˙ 2 → X1 +X ˙ 2. 6), respectively. Note that M = X1 + ˙ 2 is an invariant subspace for A× , and that M subspace for A, that M × = {0} +X and M × match in the sense that the state space of the product realization is the direct sum of M and M × .

The chapter consists of six sections. 1 presents preliminaries on realization, including the relevant deﬁnitions and the connection with systems theory. In the next two sections the realization problem is discussed. 3. 6). 1 Preliminaries on realization Let W be a rational matrix function which is also proper, that is, W has no pole at inﬁnity. As is well-known such a function can always be represented (see the next section for an explicit construction) in the form W (λ) = D + C(λI − A)−1 B. 1) Here λ is a complex variable, A is a square matrix, I is the identity matrix of the same size as A, and B and C are matrices of appropriate sizes.