# Definition The orthogonal group is the [[group]] of all real $n \times n$ [[Orthogonal Operator|orthogonal matrices]]. This [[Lie group]] has [[dimension]] $n(n-1)/2$. It is a [[Connected Group|disconnected]] group with two components, where the [[Identity Component]] being the [[Special Orthogonal Group]] $SO(n)$. See, e.g., [[Orthogonal Group in 3 Dimensions|O(3)]] for a discussion of this group in 3 dimensions. $O(n)$ is an [[Isometry Group]] over real [[vector space|vector spaces]] equipped with an [[inner product]]. See [[Isometry Group#Example 1 O n|here]] for a discussion. # Lie Algebra $\mathfrak{o}(n)$ Since $O(n) \cong \text{Isom}(V)$, element $X$ of its [[Lie Algebra]] $\mathfrak{o}(n)$ must obey: $ \braket{X \pmb{v} | \pmb{w}} = - \braket{\pmb{v}|X\pmb{w}} \quad \forall \pmb{v}, \pmb{w} \in V $ Now take $V = \mathbb{R}^n$ with the Euclidean [[inner product|dot product]], and pick an [[orthogonality|orthonormal]] [[basis]]. Then we have: $ \begin{align} (X [\pmb{v}])^T [\pmb{w}] &= -[\pmb{v}]^T X [\pmb{w}]\\ [\pmb{v}]^T X^T [\pmb{w}] &= -[\pmb{v}]^T X [\pmb{w}]\\ X^T &= -X \end{align} $ Thus, $\mathfrak{o}(n)$ is the set of $n\times n$ [[skew-symmetric matrix|antisymmetric matrices]]. %% $ \mathfrak{o}(n) = \{X \in M_n(\mathbb{R}) | X = - X^T\} $ %% The Lie algebra has the same dimension as its Lie group, $n(n-1)/2$, as one can easily verify. Note that $\mathfrak{o}(n) = \mathfrak{so}(n)$, the Lie algebra of the [[Special Orthogonal Group]]. See [[Special Orthogonal Group#Lie Algebra|here]] for a discussion.