Category:Definitions/Orthogonal Matrices
Jump to navigation
Jump to search
This category contains definitions related to Orthogonal Matrices.
Related results can be found in Category:Orthogonal Matrices.
Let $R$ be a ring with unity.
Let $\mathbf Q$ be an invertible 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:
Pages in category "Definitions/Orthogonal Matrices"
The following 4 pages are in this category, out of 4 total.