(See the appendix for the notion of equivalence.) In this case C is said to be a linearization of L(λ). It will be seen that there are ln × ln matrices C such that andλI−C are equivalent. We call the matrix-valued function L (λ) = Iλl + Σ j=0 l−1 Aj λj a monic matrix polynomial of degree l. Let A0, A1, …, Al−1 be complex n × n matrices. Let J1 and J2 be matrices in Jordan normal form with sizes p × p and q × q, respectively. These spectral properties ensure that the partial multiplicities corresponding to a particular eigenvalue λ0 are the same for A0 and its extension A. We start with a relatively simple but important case in which A as well as its extension are in the Jordan form and have special spectral properties. We are interested in the Jordan form (or, equivalently, the partial multiplicities) of A0 and its extensions. Also, A0 is called the restriction of A to ℳ. A linear transformation A: C/ n → C/ n is called an extension of A0 if Ax= A0 x for every x ∈ ℳ. Let ℳ⊂ C/ n be a subspace, and consider a transformation A0 :ℳ→ℳ. Here, we present partial results and important inequalities.Ĥ.1 EXTENSIONS FROM AN INVARIANT SUBSPACE The main problems of this chapter are: given Jordan normal forms for A|N and PA|ℳ, what are the possible Jordan forms for A itself? In general, this problem is open. Thus there is an A-invariant subspace N such that C/ n =ℳ∔N and there is a projector P onto ℳ along N. Ĭonsider a transformation A: C/ n → C/ n and an A-coinvariant subspace ℳ. Trivial examples of invariant subspaces are and C/ n.
In other words, ℳ is invariant for A means that the image of ℳ under A is contained in ℳ Aℳ ⊂ ℳ. A subspace ℳ⊂ C/ n is called invariant for the transformation A, or A invariant, if Ax ∈ ℳ for every vector x ∈ ℳ. Let A: C/ n → C/ n be a linear transformation. The presentation of the material here is elementary and does not even require use of the Jordan form. The lattice of invariant subspaces of a linear transformation-a notion that will be important in the sequel-is introduced. We also study the behaviour of invariant subspaces of a transformation when the operations of similarity and taking adjoints are applied to the transformation. Some basic tools (projectors, factor spaces, angular transformations, triangular forms) for the study of invariant subspaces are developed.
It contains the simplest properties of invariant subspaces of a linear transformation. The latter are not dealt with separately, but are integrated into the text in a way that is natural in the development of the mathematical structure. In this book the reader will find a treatment of certain aspects of linear algebra that meets the two objectives: to develop systematically the central role of invariant subspaces in the analysis of linear transformations and to include relevant recent developments of linear algebra stimulated by linear systems theory. As examples of new concepts of linear algebra developed to meet the needs of systems theory, we should mention invariant subspaces for nonsquare matrices and similarity of such matrices. In the treatment of such problems new concepts and theories have been developed that form complete new chapters in the body of linear algebra. The need for such a treatment has become more apparent in recent years because of developments in different fields of application and especially in linear systems theory, where concepts such as controllability, feedback, factorization, and realization of matrix functions are commonplace. It seems to the authors that now there is a case for developing a treatment of linear algebra in which the central role of invariant subspace is systematically followed up. Probably for this reason, the first books on invariant subspaces appeared in the framework of infinite-dimensional spaces. Here, it can be argued that the structure is poorer and this is one of the few available tools for the study of many classes of operators. The importance of invariant subspaces becomes clearer in the context of operator theory on spaces of infinite dimension. In particular, the notion of an invariant subspace as an entity is often lost in the discussion of eigenvalues, eigenvectors, generalized eigenvectors, and so on. Perhaps because the whole structure is very rich, the treatment becomes fragmented as other related ideas and notions intervene. However, in existing texts and expositions the notion is not easily or systematically followed. Invariant subspaces are a central notion of linear algebra.
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