INDEFINITE LINEAR ALGEBRA AND APPLICATIONS
CONTENTS INTRODUCTION AND OUTLINE DESCRIPTION OF THE CONTENTS NOTATION AND CONVENTIONS INDE?NITE INNER PRODUCTS DE?NITION ORTHOGONALITY AND ORTHOGONAL BASES CLASSI?CATION OF SUBSPACES EXERCISES NOTES ORTHOGONALIZATION AND ORTHOGONAL POLYNOMIALS REGULAR ORTHOGONALIZATIONS THE THEOREMS OF SZEG?O AND KREIN ONE-STEP THEOREM DETERMINANTS OF ONE-STEP COMPLETIONS CLASSES OF LINEAR TRANSFORMATIONS ADJOINT MATRICES H-SELFADJOINT MATRICES: EXAMPLES AND SIMPLEST PROPERTIES H-UNITARY MATRICES: EXAMPLES AND SIMPLEST PROPERTIES A SECOND CHARACTERIZATION OF H-UNITARY MATRICES UNITARY SIMILARITY CONTRACTIONS DISSIPATIVE MATRICES SYMPLECTIC MATRICES 4 10 NOTES CANONICAL FORMS DESCRIPTION OF A CANONICAL FORM FIRST APPLICATION OF THE CANONICAL FORM PROOF OF THEOREM 5 1 1 CLASSI?CATION OF MATRICES BY UNITARY SIMILARITY SIGNATURE MATRICES STRUCTURE OF H-SELFADJOINT MATRICES H-DE?NITE MATRICES SECOND DESCRIPTION OF THE SIGN CHARACTERISTIC STABILITY OF THE SIGN CHARACTERISTIC 5 10 CANONICAL FORMS FOR PAIRS OF HERMITIAN MATRICES 5 11 THIRD DESCRIPTION OF THE SIGN CHARACTERISTIC 5 12 INVARIANT MAXIMAL NONNEGATIVE SUBSPACES 5 13 INVERSE PROBLEMS 5 14 CANONICAL FORMS FOR H-UNITARIES: FIRST EXAMPLES 5 15 CANONICAL FORMS FOR H-UNITARIES: GENERAL CASE 5 16 FIRST APPLICATIONS OF THE CANONICAL FORM OF H-UNITARIES 5 17 FURTHER DEDUCTIONS FROM THE CANONICAL FORM 5 18 EXERCISES 5 19 NOTES REAL H-SELFADJOINT MATRICES REAL H-SELFADJOINT MATRICES AND CANONICAL FORMS PROOF OF THEOREM 6 1 5 COMPARISON WITH RESULTS IN THE COMPLEX CASE CONNECTED COMPONENTS OF REAL UNITARY SIMILARITY CLASSES FUNCTIONS OF H-SELFADJOINT MATRICES PRELIMINARIES EXPONENTIAL AND LOGARITHMIC FUNCTIONS THE CANONICAL FORM AND SIGN CHARACTERISTIC FUNCTIONS WHICH ARE SELFADJOINT IN ANOTHER INDE?NITE INNER PRODUCT H-NORMAL MATRICES DECOMPOSABILITY: FIRST REMARKS ON UNITARY SIMILARITY IN AN INDE?NITE INNER PRODUCT THE CASE OF ONLY ONE NEGATIVE EIGENVALUE OF H GENERAL PERTURBATIONS STABILITY OF DIAGONALIZABLE MATRICES GENERAL PERTURBATIONS OF H-SELFADJOINT MATRICES STABLY DIAGONALIZABLE H-SELFADJOINT MATRICES ANALYTIC PERTURBATIONS AND EIGENVALUES ANALYTIC PERTURBATIONS AND EIGENVECTORS THE REAL CASE POSITIVE PERTURBATIONS OF H-SELFADJOINT MATRICES H-SELFADJOINT STABLY R-DIAGONALIZABLE MATRICES H-UNITARILY STABLY U-DIAGONALIZABLE MATRICES 9 10 EXERCISES 9 11 NOTES 10 DE?NITE INVARIANT SUBSPACES 10 1 SEMIDE?NITE AND NEUTRAL SUBSPACES: A PARTICULAR H 10 2 PLUS MATRICES AND INVARIANT NONNEGATIVE SUBSPACES 10 3 DEDUCTIONS FROM THEOREM 10 2 4 10 4 EXPANSIVE, CONTRACTIVE MATRICES AND SPECTRAL PROPERTIES 10 5 THE REAL CASE 10 6 EXERCISES 10 7 NOTES 11 DI?ERENTIAL EQUATIONS OF FIRST ORDER 11 1 BOUNDEDNESS OF SOLUTIONS 11 2 HAMILTONIAN SYSTEMS OF POSITIVE TYPE WITH CONSTANT COE?CIENTS 11 3 EXERCISES 11 4 NOTES 12 MATRIX POLYNOMIALS 12 1 STANDARD PAIRS AND TRIPLES 12 2 MATRIX POLYNOMIALS WITH HERMITIAN COE?CIENTS 12 3 FACTORIZATION OF HERMITIAN MATRIX POLYNOMIALS 12 4 THE SIGN CHARACTERISTIC OF HERMITIAN MATRIX POLYNOMIALS 12 5 THE SIGN CHARACTERISTIC OF HERMITIAN ANALYTIC MATRIX FUNCTIONS 12 6 HERMITIAN MATRIX POLYNOMIALS ON THE UNIT CIRCLE 12 7 EXERCISES 12 8 NOTES 13 DI?ERENTIAL AND DI?ERENCE EQUATIONS OF HIGHER ORDER 13 1 GENERAL SOLUTION OF A SYSTEM OF DI?ERENTIAL EQUATIONS 13 2 BOUNDEDNESS FOR A SYSTEM OF DI?ERENTIAL EQUATIONS 13 3 STABLE BOUNDEDNESS FOR DI?ERENTIAL EQUATIONS 13 4 THE STRONGLY HYPERBOLIC CASE 13 5 CONNECTED COMPONENTS OF DI?ERENTIAL EQUATIONS 13 6 A SPECIAL CASE 13 7 DI?ERENCE EQUATIONS 13 8 STABLE BOUNDEDNESS FOR DI?ERENCE EQUATIONS 13 9 CONNECTED COMPONENTS OF DI?ERENCE EQUATIONS 13 10EXERCISES 13 11NOTES 14 ALGEBRAIC RICCATI EQUATIONS 14 1 MATRIX PAIRS IN SYSTEMS THEORY AND CONTROL 14 2 ORIGINS IN SYSTEMS THEORY 14 3 PRELIMINARIES ON THE RICCATI EQUATION 14 4 SOLUTIONS AND INVARIANT SUBSPACES 14 5 SYMMETRIC EQUATIONS 14 6 AN EXISTENCE THEOREM 14 7 EXISTENCE WHEN M HAS REAL EIGENVALUES 14 8 DESCRIPTION OF HERMITIAN SOLUTIONS 14 9 EXTREMAL HERMITIAN SOLUTIONS 14 10THE CARE WITH REAL COE?CIENTS 14 11THE CONCERNS OF NUMERICAL ANALYSIS 14 12EXERCISES 14 13NOTES BIBLIOGRAPHY