Orthogonal matrix

A real square matrix is said to be orthogonal if all of its rows or columns are orthogonal vectors. Further, the matrix is said to be orthonormal if all of its rows or columns are orthonormal vectors.

Characterizations#

Any one of the following properties is sufficient to identify an orthogonal matrix $\mathrm{A}$ and are all true for such a matrix:

• $\mathrm{A}^{T}\mathrm{A}=\mathrm{A}\mathrm{A}^{T}=\mathrm{I}$ where $^{T}$ denotes transposition and $\mathrm{I}$ is the identity matrix.
• $\mathrm{A}^{T}=\mathrm{A}^{-1}$ where $\mathrm{A}^{-1}$ is the inverse matrix.

Properties#

• It is invertible with inverse $\mathrm{A}^{-1}=\mathrm{A}^{T}$.
• It is Identity matrix $\mathrm{A}^{-1}=\mathrm{A}^{*}$ where $\mathrm{A}^{*}$ denotes the conjugate transpose.
• The determinant $\det \mathrm{A}$ is either $+1$ or $-1$.