Computational matrix representation modules for linear operators with explicit constructions for a class of Lie operators. (English) Zbl 0932.65052
The authors develop a procedure for constructing matrix representations for a class of linear operators on finite-dimensional spaces. First serial number functions for locating basis monomials in the linear space of homogeneous polynomials of fixed degree, ordered under structural lexicographic are presented. Next basic lemmas describing the modular structure of matrix representations for operators constructed conically from elementary operators are presented.
Using these results, explicit matrix representations are then given for the Lie derivative and Lie-Poisson bracket operators defined on spaces of homogeneous polynomials. In particular, they are comprized of blocks obtained as Kronecker sums of modular components, each corresponding to specific Jordan blocks. At an implementation level, recursive programming is applied to construct these modular components explicitly.
The results are also applied to computing power series approximations for the center manifold of a dynamical system. In this setting, the linear operator of interest is parameterized by two matrices, a generalization of the Lie-Poisson bracket.
Using these results, explicit matrix representations are then given for the Lie derivative and Lie-Poisson bracket operators defined on spaces of homogeneous polynomials. In particular, they are comprized of blocks obtained as Kronecker sums of modular components, each corresponding to specific Jordan blocks. At an implementation level, recursive programming is applied to construct these modular components explicitly.
The results are also applied to computing power series approximations for the center manifold of a dynamical system. In this setting, the linear operator of interest is parameterized by two matrices, a generalization of the Lie-Poisson bracket.
Reviewer: P.Narain (Bombay)
MSC:
| 65F30 | Other matrix algorithms (MSC2010) |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
Keywords:
matrix representations; linear operators; Lie derivative; Lie-Poisson bracket operators; Jordan blocks; recursive programming; center manifold; dynamical systemReferences:
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