Compatible Lie und Jordan algebras and applications to structured matrices and pencils

Christian Mehl

ISBN 978-3-89722-173-4
109 pages, year of publication: 1999
price: 40.00 €

Abstract: In the thesis the theory of compatible Lie and Jordan algebras is discussed and applied to structured matrices and matrix pencils. The theory of compatible Lie and Jordan algebras deals with the abstract relation of some Lie and Jordan subalgebras of an associative algebra, in particular the relation of the Lie algebra of K-skew-Hermitian matrices and the Jordan algebra of K-Hermitian matrices, i.e., the algebras of matrices that are (skew-)Hermitian with respect to the indefinite quadratic form defined by a nonsingular matrix K. Furthermore, the structure preserving transformations of these algebras and the corresponding Lie groups are characterized. This theory and the theory of K-unitary, K-skew-Hermitian and K-Hermitian pencils provide a basis for the discussion of the structure of Riccati pencils that arise in the linear quadratic optimal control problem or the theory of the generalized algebraic Riccati equations. This discussion leads to the theory of skew-Hamiltonian/Hamiltonian pencils that canonical and Schur-type forms for these kind of pencils are presented in. Furthermore Schur-like forms are presented for double-structured matrices that arise in quantum chemistry .