A term loosely applying to groups of matrices under matrix multiplication, particularly important in geometry. Examples include:
GL(n,F) | The general linear group. Invertible n × n matrices with entries in the field F. |
SL(n,F) | The special linear group. Determinant 1 n × n matrices with entries in F. |
O(n) | orthogonal n × n matrices. |
SO(n) | orthogonal n × n matrices with determinant 1. |
U(n) | unitary n × n matrices. |
SU(n) | unitary n × n matrices with determinant 1. |
AGL(n,F) | affine transformations on Fn. |
PGL(n,F) | projective transformations on n–1 dimensional projective space over F. |
Lorentz group | the linear isometries (see linear map) of Minkowski space. |
Sp(2n,F) | sympletic group of those 2n × 2n matrices M over F satisfying MTΩM = Ω, where |
See also Lie groups, linear groups.