GL(n,R) - Knowlpedia

For every integer $n\ge1$ , $\operatorname{GL}(n,\mathbb R)$ is the general linear group

\operatorname{GL}(n,\mathbb R)=\{A\in M_{n}(\mathbb R):\det A\ne0\},

with matrix multiplication and its usual real smooth manifold structure.

Dimensions and structure

This group has real dimension $n^2$ . It is open in the space of all matrices. Its tangent algebra consists of all matrices with the commutator bracket. The sign of the determinant distinguishes its two connected components.

The projectivization map has kernel all nonzero scalar matrices. See $\operatorname{PGL}(n,\mathbb R)$ .

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.