Partially specified matrices and operators: classification, completion, applications
Birkhäuser Verlag, 1995 - Mathematics - 333 pages
This book explores a new direction in linear algebra and operator theory dealing with the invariants of partially specified matrices and operators, and with the spectral analysis of their completions. The theory developed centers around two major problems concerning matrices of which part of the entries are given and the others are unspecified. The first is a problem of classification of partially specified matrices, and the results here may be seen as a far reaching generalization of the Jordan canonical form. The second problem is the eigenvalue completion problem, which asks for a description of all possible eigenvalues and their multiplicities of the matrices which one obtains by filling in the unspecified entries. Both problems are also considered in an infinite dimensional framework. A large part of the book deals with applications to matrix theory and analysis, namely to stabilization problems in mathematical system theory, to problems of Wiener-Hopf factorization and interpolation for matrix polynomials and rational matrix functions, to the Kronecker structure theory of linear pencils, and to non-everywhere defined operators. The book will appeal to a wide group of mathematicians and engineers. Much of the material can be used in advanced courses in matrix and operator theory.
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Elementary operations on blocks
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admissible similarity analytic continuation assume Banach space basis block associated block B;P,Q block similarity bounded linear operators called canonical form chapter compute condition Corollary decomposable diagonal dim Ker direct sum dual sequence eigenvalue completion problem entries equal finite dimensional finite type follows full length block full range pair full width block given hence invariant polynomials invariant subspace invertible matrix invertible operator Jordan blocks KerP KerQ kind with index Lemma linear operator linear space linear transformation matrix polynomial modulo monic polynomial non-everywhere defined operator numbers operator block operator pencils positioned submatrix projection proof of Theorem Proposition prove rank rational matrix function respect result right Jordan pair right null pair root functions Schagen second kind strictly equivalent subspace T-null pair Theorem 2.1 third kind triple U-similar U-specified matrix underlying space vectors Wiener-Hopf factorization zero kernel pair