Category:Orthogonal Matrices

This category contains results about Orthogonal Matrices.
Definitions specific to this category can be found in Definitions/Orthogonal Matrices.

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 $\mathbf Q^\intercal$.