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

### Christian Mehl

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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 .