Definition:Orthogonal Group
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This page is about Orthogonal Group. For other uses, see Orthogonal.
Definition
Let $k$ be a field.
The ($n$th) orthogonal group (on $k$), denoted $\map {\mathrm O} {n, k}$, is the following subset of the general linear group $\GL {n, k}$:
- $\map {\mathrm O} {n, k} := \set {M \in \GL {n, k}: M^\intercal = M^{-1} }$
where $M^\intercal$ denotes the transpose of $M$.
Further, $\map {\mathrm O} {n, k}$ is considered to be endowed with conventional matrix multiplication.
That is, the ($n$th) orthogonal group (on $k$) is the set of all orthogonal order-$n$ square matrices over $k$ under (conventional) matrix multiplication.
Orthogonal Group of Bilinear Form
Let $V$ be a vector space over a field $\mathbb K$.
Let $B: V \times V \to \mathbb K$ be a nondegenerate bilinear form.
Its orthogonal group $\map {\mathrm O} B$ is the group of invertible linear transformations $g \in \GL V$ such that:
- $\forall v, w \in V : \map B {g v, g w} = \map B {v, w}$
Orthogonal Group of Inner Product Space
Let $V$ be an inner product space.
Its orthogonal group $\map {\mathrm O} V$ is the group of invertible linear transformations $g \in \GL V$ such that:
- $\forall v, w \in V: \innerprod {g v} {g w} = \innerprod v w$
That is, it is the orthogonal group of its inner product.
Also see
- Definition:Special Orthogonal Group
- Definition:Unitary Group
- Orthogonal Group is Group
- Orthogonal Group is Subgroup of General Linear Group
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 3$: Examples of Infinite Groups: $\text{(iv) (b)}$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): orthogonal group
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 7$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): orthogonal group