$\mathrm{O}(n)$ is a [[Matrix Lie group]] that is the subgroup of [[GL(n;F)]] consisting of [orthogonal matrices](Orthogonal%20operators.md#Orthogonal%20matrices). Matrices in this group give rise to rotations or combinations of rotations and reflections of vectors in $\mathbb{R}$. # Corresponding Lie Algebra to $\mathrm{O}(n)$ The [[Lie algebras]], $\mathfrak{o}(n)$ consists of all real matrices $X$ satisfying $X^T = -X$, which is needed to generate of the inverse of the [Matrix exponentials](Matrix%20exponentials.md) in the [[Lie groups]]. #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups