Definition:Orthogonal Matrix
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This page is about orthogonal matrix. For other uses, see orthogonal.
Definition
Let $R$ be a ring with unity.
Let $\mathbf Q$ be a nonsingular square matrix over $R$.
Definition 1
Then $\mathbf Q$ is orthogonal if and only if:
- $\mathbf Q^{-1} = \mathbf Q^\intercal$
where:
- $\mathbf Q^{-1}$ is the inverse of $\mathbf Q$
- $\mathbf Q^\intercal$ is the transpose of $\mathbf Q$
Definition 2
Then $\mathbf Q$ is orthogonal if and only if:
- $\mathbf Q^\intercal \mathbf Q = \mathbf I$
where:
- $\mathbf Q^\intercal$ is the transpose of $\mathbf Q$
- $\mathbf I$ is the identity matrix of the same order as $\mathbf Q$.
Definition 3
Then $\mathbf Q$ is orthogonal if and only if:
- $\mathbf Q = \paren {\mathbf Q^\intercal}^{-1}$
where:
- $\mathbf Q^\intercal$ is the transpose of $\mathbf Q$
- $\paren {\mathbf Q^\intercal}^{-1}$ is the inverse of $\mathbf Q^\intercal$.
Examples
$2 \times 2$ Rotation Matrix
The rotation matrix for a plane rotation about the origin through an angle $\theta$:
- $\mathbf P = \begin {pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end {pmatrix}$
is an orthogonal matrix.
$2 \times 2$ Reflection Matrix
The reflection matrix for a plane reflection in the line $y = x \tan \dfrac \theta 2$:
- $\mathbf Q = \begin {pmatrix} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end {pmatrix}$
is an orthogonal matrix.
Also see
- Results about orthogonal matrices can be found here.