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.