general linear group
English
Noun
general linear group (plural general linear groups)
- (group theory) For given field F and order n, the group of invertible n×n matrices, with the group operation of matrix multiplication.
- 1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, Springer, 2000, Softcover Reprint, page 17,
- Often Lie groups arise as subgroups of certain larger Lie groups; for example, the orthogonal groups are subgroups of the general linear groups of all invertible matrices.
- 2003, Vladislav V. Goldberg (translator), Maks Aizikovich Akivis, Tensor Calculus with Applications, World Scientific, page 119,
- We will again call this group the general linear group and denote it by GL3.
- In just the same way, the set of all nonsingular linear transformations of the plane L2 is a group denoted by GL2 and called the general linear group of order two.
- 2009, Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants, Springer, page 1,
- We show how to put a Lie group structure on a closed subgroup of the general linear group and determine the Lie algebras of the classical groups.
- 1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, Springer, 2000, Softcover Reprint, page 17,
Usage notes
The general linear group can be denoted GL(n, F) or GLn(F) — or, if the field is understood, GL(n) or GLn.
In the cases that F is the field of the real or of the complex numbers, GL(n, F) is a Lie group.
Derived terms
Related terms
Translations
group of invertible n×n matrices
See also
Further reading
General semilinear group on Wikipedia.Wikipedia
Matrix group on Wikipedia.Wikipedia
Projective linear group on Wikipedia.Wikipedia