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Bilinear and Sesquilinear Forms
New Features:
- A polar space, namely a vector space with an attached bilinear, sesquilinear
or quadratic form, can be constructed from a standard form or from a
user-supplied form. The type of a polar space is one of: symplectic,
pseudo-symplectic, unitary, quadratic or orthogonal, according to whether
the form is alternating, pseudo-alternating, hermitian, quadratic or
symmetric.
- New functions IsometryGroup(V) and SimilarityGroup(V) return
the group of all isometries and the group of all similarities of the polar
space V.
- Given polar spaces V and W, the function IsIsometric(V,W)
determines whether there is an isometry from V to W and, if
so, returns it.
- The function PseudoSymplecticGroup(n,q) returns the pseudo-symplectic
group of n x n
matrices over the field
GF(q)
(where q
must be a
power of 2
).
- The function LieAlgebraFromForm(J) returns the Lie algebra of
derivations of the form J.
- The functions HyperbolicPair, Witt Decomposition and
ExtendIsometry have been enhanced to work with all types of polar
spaces. In particular, Witt's Theorem (using ExtendIsometry) is now
available for unitary spaces.
- Given a matrix group G
there are new functions to compute the bilinear,
sesquilinear and quadratic forms which are invariant (or invariant up to
a scalar multiple) under the action of G
. If G
(and its derived group)
is absolutely irreducible this functionality has previously been available
via ClassicalForms(G).
Next: Linear Associative Algebras
Up: Linear Algebra and Module
Previous: Sparse Matrices