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Matrix Lie Group

Matrix Lie Groups

Matrix Lie Group
Notation
Field
Property
General Linear Group
GL(n)
C or R
n×n invertible matrices
Special Linear Group
SL(n)
C or R
GL(n), det⁡=1
Unitary Group
U(n)
C
GL(n), U^∗=U^(-1)
Special Unitary Group
SU(n)
C
GL(n), U^∗=U^(-1), det⁡=1
Orthogonal Group
O(n)
R
GL(n), R^⊤=R^(-1)
Special Orthogonal Group
SO(n)
R
GL(n), R^⊤=R^(-1), det⁡=1
Euclidean Group
E(n)
R
GL(n+1), T= [R p; 0 1]
Special Euclidean Group
SE(n)
R
GL(n+1), T= [R p; 0 1], det⁡=1
  • Notes: A matrix Lie group is a closed subgroup of GL(n, C). The common binary operation is matrix multiplication.
  • Please see the textbook “Lie Groups, Lie Algebras, and Representations” by Brian C. Hall for more details.