Cover -- Contents -- Preface -- 1 Introduction -- 1.1 Notation and conventions -- 1.2 Standard matrices -- 2 The algebra of quaternions -- 2.1 Basic definitions and properties -- 2.2 Real linear transformations and equations -- 2.3 The Sylvester equation -- 2.4 Automorphisms and involutions -- 2.5 Quadratic maps -- 2.6 Real and complex matrix representations -- 2.7 Exercises -- 2.8 Notes -- 3 Vector spaces and matrices: Basic theory -- 3.1 Finite dimensional quaternion vector spaces -- 3.2 Matrix algebra -- 3.3 Real matrix representation of quaternions -- 3.4 Complex matrix representation of quaternions -- 3.5 Numerical ranges with respect to conjugation -- 3.6 Matrix decompositions: nonstandard involutions -- 3.7 Numerical ranges with respect to nonstandard involutions -- 3.8 Proof of Theorem 3.7.5 -- 3.9 The metric space of subspaces -- 3.10 Appendix: Multivariable real analysis -- 3.11 Exercises -- 3.12 Notes -- 4 Symmetric matrices and congruence -- 4.1 Canonical forms under congruence -- 4.2 Neutral and semidefinite subspaces -- 4.3 Proof of Theorem 4.2.6 -- 4.4 Proof of Theorem 4.2.7 -- 4.5 Representation of semidefinite subspaces -- 4.6 Exercises -- 4.7 Notes -- 5 Invariant subspaces and Jordan form -- 5.1 Root subspaces -- 5.2 Root subspaces and matrix representations -- 5.3 Eigenvalues and eigenvectors -- 5.4 Some properties of Jordan blocks -- 5.5 Jordan form -- 5.6 Proof of Theorem 5.5.3 -- 5.7 Jordan forms of matrix representations -- 5.8 Comparison with real and complex similarity -- 5.9 Determinants -- 5.10 Determinants based on real matrix representations -- 5.11 Linear matrix equations -- 5.12 Companion matrices and polynomial equations -- 5.13 Eigenvalues of hermitian matrices -- 5.14 Differential and difference equations -- 5.15 Appendix: Continuous roots of polynomials -- 5.16 Exercises -- 5.17 Notes.

6 Invariant neutral and semidefinite subspaces -- 6.1 Structured matrices and invariant neutral subspaces -- 6.2 Invariant semidefinite subspaces respecting conjugation -- 6.3 Proof of Theorem 6.2.6 -- 6.4 Unitary, dissipative, and expansive matrices -- 6.5 Invariant semidefinite subspaces: Nonstandard involution -- 6.6 Appendix: Convex sets -- 6.7 Exercises -- 6.8 Notes -- 7 Smith form and Kronecker canonical form -- 7.1 Matrix polynomials with quaternion coefficients -- 7.2 Nonuniqueness of the Smith form -- 7.3 Statement of the Kronecker form -- 7.4 Proof of Theorem 7.3.2: Existence -- 7.5 Proof of Theorem 7.3.2: Uniqueness -- 7.6 Comparison with real and complex strict equivalence -- 7.7 Exercises -- 7.8 Notes -- 8 Pencils of hermitian matrices -- 8.1 Canonical forms -- 8.2 Proof of Theorem 8.1.2 -- 8.3 Positive semidefinite linear combinations -- 8.4 Proof of Theorem 8.3.3 -- 8.5 Comparison with real and complex congruence -- 8.6 Expansive and plus-matrices: Singular H -- 8.7 Exercises -- 8.8 Notes -- 9 Skewhermitian and mixed pencils -- 9.1 Canonical forms for skewhermitian matrix pencils -- 9.2 Comparison with real and complex skewhermitian pencils -- 9.3 Canonical forms for mixed pencils: Strict equivalence -- 9.4 Canonical forms for mixed pencils: Congruence -- 9.5 Proof of Theorem 9.4.1: Existence -- 9.6 Proof of Theorem 9.4.1: Uniqueness -- 9.7 Comparison with real and complex pencils: Strict equivalence -- 9.8 Comparison with complex pencils: Congruence -- 9.9 Proofs of Theorems 9.7.2 and 9.8.1 -- 9.10 Canonical forms for matrices under congruence -- 9.11 Exercises -- 9.12 Notes -- 10 Indefinite inner products: Conjugation -- 10.1 H-hermitian and H-skewhermitian matrices -- 10.2 Invariant semidefinite subspaces -- 10.3 Invariant Lagrangian subspaces I -- 10.4 Differential equations I.

10.5 Hamiltonian, skew-Hamiltonian matrices: Canonical forms -- 10.6 Invariant Lagrangian subspaces II -- 10.7 Extension of subspaces -- 10.8 Proofs of Theorems 10.7.2 and 10.7.5 -- 10.9 Differential equations II -- 10.10 Exercises -- 10.11 Notes -- 11 Matrix pencils with symmetries: Nonstandard involution -- 11.1 Canonical forms for ø-hermitian pencils -- 11.2 Canonical forms for ø-skewhermitian pencils -- 11.3 Proof of Theorem 11.2.2 -- 11.4 Numerical ranges and cones -- 11.5 Exercises -- 11.6 Notes -- 12 Mixed matrix pencils: Nonstandard involutions -- 12.1 Canonical forms for ø-mixed pencils: Strict equivalence -- 12.2 Proof of Theorem 12.1.2 -- 12.3 Canonical forms of ø-mixed pencils: Congruence -- 12.4 Proof of Theorem 12.3.1 -- 12.5 Strict equivalence versus ø-congruence -- 12.6 Canonical forms of matrices under ø-congruence -- 12.7 Comparison with real and complex matrices -- 12.8 Proof of Theorem 12.7.4 -- 12.9 Exercises -- 12.10 Notes -- 13 Indefinite inner products: Nonstandard involution -- 13.1 Canonical forms: Symmetric inner products -- 13.2 Canonical forms: Skewsymmetric inner products -- 13.3 Extension of invariant semidefinite subspaces -- 13.4 Proofs of Theorems 13.3.3 and 13.3.4 -- 13.5 Invariant Lagrangian subspaces -- 13.6 Boundedness of solutions of differential equations -- 13.7 Exercises -- 13.8 Notes -- 14 Matrix equations -- 14.1 Polynomial equations -- 14.2 Bilateral quadratic equations -- 14.3 Algebraic Riccati equations -- 14.4 Exercises -- 14.5 Notes -- 15 Appendix: Real and complex canonical forms -- 15.1 Jordan and Kronecker canonical forms -- 15.2 Real matrix pencils with symmetries -- 15.3 Complex matrix pencils with symmetries -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V.