Parametrization of (C,A)-invariant subspaces

✍ Scribed by D. Hinrichsen; H.F. Münzner; D. Prätzel-Wolters

• Publisher: Elsevier Science
• Year: 1981
• Tongue: English
• Weight: 485 KB
• Volume: 1
• Category: Article
• ISSN: 0167-6911
• DOI: 10.1016/s0167-6911(81)80035-7

✦ Synopsis

In this paper we classify and parametrize the (C. A)-invariant subspaces of a given observable pair (C.A). For that purpose we associate with every (C.A)-invariant subspace V a K[s]module MY of Laurent series in s-' and parametrize these modules via their uniquely determined Kronecker-Hermite bases. The discrete and continuous parameters thus obtained are interpreted in state space terms. Finally we illustrate the results by presenting a complete overview over the set of all (C.A)-invariant subspaces for a given pair (C'.A)ER~*'~+~' in dual Brunovsky canonical form. 1.1. Proposition. (i) A subspace V of X is (C, A)invariant if and only if A(VflKerC)CV. (1.2)

(ii) For ever:y subspace E of X there exists a unique smallest (C, A)-invariant subspace V,( E) containing E.

(iii) The set :7i, Aj of ull (C, A