general linear group
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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, Maks Aizikovich Akivis, translated by Vladislav V. Goldberg, 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
