# 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. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://wiki.hanzheteng.com/math/algebra/matrix-lie-group.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.