Polar decomposition and Brion’s theorem
نویسندگان
چکیده
In this note we point out the relation between Brion’s formula for the lattice point generating function of a convex polytope in terms of the vertex cones [Bri88] on the one hand, and the polar decomposition à la Lawrence/Varchenko [Law91a, Var87] on the other. We then go on to prove a version of polar decomposition for non-simple polytopes.
منابع مشابه
Finding the polar decomposition of a matrix by an efficient iterative method
Theobjective in this paper to study and present a new iterative method possessing high convergence order for calculating the polar decompostion of a matrix. To do this, it is shown that the new scheme is convergent and has high convergence. The analytical results are upheld via numerical simulations and comparisons.
متن کاملThe interplay of the polar decomposition theorem and the Lorentz group
Abstract: It is shown that the polar decomposition theorem of operators in (real) Hilbert spaces gives rise to the known decomposition in boost and spatial rotation part of any matrix of the orthochronous proper Lorentz group SO(1, 3)↑. This result is not trivial because the polar decomposition theorem is referred to a positive defined scalar product while the Lorentz-group decomposition theore...
متن کاملBeck ,
We discuss and give elementary proofs of results of Brion and of LawrenceVarchenko on the lattice-point enumerator generating functions for polytopes and cones. While this note is purely expository, its contains a new proof of Brion’s Theorem using irrational decompositions.
متن کاملSymbolic computation of the Duggal transform
Following the results of cite{Med}, regarding the Aluthge transform of polynomial matrices, the symbolic computation of the Duggal transform of a polynomial matrix $A$ is developed in this paper, using the polar decomposition and the singular value decomposition of $A$. Thereat, the polynomial singular value decomposition method is utilized, which is an iterative algorithm with numerical charac...
متن کاملAdjoints, absolute values and polar decompositions
Various questions about adjoints, absolute values and polar decompositions of operators are addressed from a constructive point of view. The focus is on bilinear forms. Conditions are given for the existence of an adjoint, and a general notion of a polar decomposition is developed. The Riesz representation theorem is proved without countable choice.
متن کامل