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 $\mathbf Q^\intercal$.


Also see

  • Results about orthogonal matrices can be found here.