general linear group

English

Noun

general linear group (plural general linear groups)

1. (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.

Usage notes

The general linear group can be denoted GL(n, F) or GL_n(F) — or, if the field is understood, GL(n) or GL_n.

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

• projective general linear group

Related terms

• linear group

Translations

group of invertible n×n matrices

See also

• Lie group
• matrix group
• special linear group

Further reading

• General semilinear group on Wikipedia.Wikipedia
• Matrix group on Wikipedia.Wikipedia
• Projective linear group on Wikipedia.Wikipedia