Partially Specified Matrices and Operators: Classification, Completion, Applications

This book is devoted to a new direction in linear algebra and operator theory that deals 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 classification problem and aims at a simplification of the given part with the help of admissible similarities. The results here may be seen as a far reaching generalization of the Jordan canonical form. The second problem is called the eigenvalue completion problem and asks to describe 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 operator 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 eigenvalue completion problem has a natural associated inverse, which appears as a restriction problem. The analysis of these two problems is often simpler when a solution of the corresponding classification problem is available.

I. Main problems and motivation.- I.1 Eigenvalue completion problems and first examples.- I.2 Reduction by similarity.- I.3 Blocks.- I.4 Block similarity.- I.5 Special cases of block similarity.- I.6 Eigenvalue completion and restriction problems.- Notes.- II. Elementary operations on blocks.- II.1 Block-invariant subspaces.- II.2 Direct sums of blocks and decomposable blocks.- II.3 Indecomposable blocks.- II.4 Duality of blocks.- Notes.- III. Full length blocks.- III.1 Structure theorems for full length blocks.- III.2 Finite dimensional operator pencils.- III.3 Similarity of non-everywhere defined linear operators.- III.4 Dual sequences.- Notes.- IV. The eigenvalue completion problem for full length blocks.- IV.1 Main theorems.- IV.2 Reduction to a problem on matrix polynomials.- IV.3 A one column completion problem for matrix polynomials.- IV.4 Proof of the first main theorem.- IV.5 Some applications of the restriction problem.- IV.6 A matrix equation.- Notes.- V. Full width blocks.- V.1 Structure theorems for full width blocks.- V.2 Finite dimensional operator pencils.- V.3 Similarity of operators modulo a subspace.- V.4 Duality.- V.5 The eigenvalue completion problem and related problems.- V.6 A matrix equation.- Notes.- VI. Principal blocks.- VI.1 Structure theorem for principal blocks.- VI.2 The eigenvalue completion problem for principal blocks.- VI.3 The eigenvalue restriction problem for principal blocks.- Notes.- VII. General blocks.- VII.1 Block similarity invariants, completion and restriction problems.- VII.2 Structure theorems and canonical form.- VII.3 Proof of Proposition 2.2.- VII.4 Proof of Theorems 1.1 and 2.1.- VII.5 Finite dimensional operator pencils.- VII.6 Non-everywhere defined operators modulo a subspace.- VII.7 Duality of operator blocks.- VII.8 The eigenvalue completion problem.- Notes.- VIII. Off-diagonal blocks.- VIII.1 Structure theorems for off-diagonal blocks.- VIII.2 The eigenvalue completion and restriction problems.- Notes.- IX. Connections with linear systems.- IX.1 Linear input/output systems and transfer functions.- IX.2 Blocks and controllability.- IX.3 Blocks and observability.- IX.4 Minimal systems.- IX.5 Feedback and block similarity.- IX.6 Eigenvalue assignment and eigenvalue completion.- IX.7 Assignment of controllability indices and eigenvalue restriction.- IX.8 (A, B)-invariant subspaces.- IX.9 Output stabilization by state feedback.- IX.10 Output injection.- Notes.- X. Applications to matrix polynomials.- X.1 Preliminaries.- X.2 Matrix polynomials with prescribed zero structure.- X.3 Wiener-Hopf factorization and indices.- Notes.- XI. Applications to rational matrix functions.- XI.1 Preliminaries on pole pairs and null pairs.- XI.2 The one sided homogeneous interpolation problem.- XI.3 Homogeneous two sided interpolation.- XI.4 An auxiliary result on block similarity.- XI.5 Factorization indices for rational matrix functions.- Notes.- XII. Infinite dimensional operator blocks.- XII.1 Preliminaries.- XII.2 Main theorems about (P, I)-blocks.- XII.3 Main theorems for (I, Q)-blocks.- XII.4 Operator blocks on a separable Hilbert space.- XII.5 Spectral completion and assignment problems.- Notes.- XIII. Factorization of operator polynomials.- XIII.1 Preliminaries on null pairs and spectral triples.- XIII.2 Wiener-Hopf equivalence.- XIII.3 Wiener-Hopf factorization.- XIII.4 Wiener-Hopf factorization and strict equivalence.- XIII.5 The Fredholm case.- Notes.- XIV. Factorization of analytic operator functions.- XIV.1 Preliminaries on spectral triples.- XIV.2 Wiener-Hopf equivalence.- XIV.3 Wiener-Hopf factorization.- Notes.- XV. Eigenvalue completion problems for triangular matrices.- XV.1 U-similar and decomposed U-specified matrices.- XV.2 Invariants for U-similarity.- XV.3 Invariants and U-canonical form in the generic case.- XV.4 The diagonal of U-similar matrices.- XV.5 An eigenvalue completion problem.- XV.6 Applications.- Notes.- Append.- A.1 Root functions of regular analytic matrix functions.- A.2 Right Jordan pairs of regular analytic matrix functions.- A.3 Left Jordan pairs.- A.4 Jordan pairs and Laurent principal parts.- A.5 Global spectral data for regular analytic matrix functions.- Notes.- List of notations.