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The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x. All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring .
Its main advantage versus a purely multigrid solver is particularly clear for nonlinear problems, e.g., eigenvalue problems. If the matrix of the original equation or an eigenvalue problem is symmetric positive definite (SPD), the preconditioner is commonly constructed to be SPD as well, so that the standard conjugate gradient (CG) iterative ...
Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic geometry. This core functionality includes arithmetic on rings, modules, and matrices, as well as algorithms for Gröbner bases, free resolutions, Hilbert series, determinants and Pfaffians, factoring, and similar.
Formally, the symmetric algebra of a vector space V over a field F is the group algebra of the dual, Sym(V) := F[V ∗], and the Weyl algebra is the group algebra of the (dual) Heisenberg group W(V) = F[H(V ∗)]. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V).
In spite of the name, the notion is a common generalization of a symmetric algebra and an exterior algebra: indeed, if V is a (non-graded) R-module, then the graded-symmetric algebra of V with trivial grading is the usual symmetric algebra of V. Similarly, the graded-symmetric algebra of the graded module with V in degree one and zero elsewhere ...
In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL(V). A field automorphism is a bijective ring homomorphism from a field to itself.
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