# 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: $\text{(iv) (b)}$ - 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**