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Matrix Lie Group
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.
Last modified 5mo ago