# Definition An orthogonal operator is one characterized by: $ O^T O = O O^T = I $ or $ O^T = O^{-1} $ They can be considered as the real version of [[Unitary Operator|unitary operators]]. # Properties Orthogonal operators preserve the [[inner product]], i.e. $\braket{O \phi | O \psi} = \braket{\phi|\psi}$ # Orthogonal Matrices [[Matrix|Matrices]] representing orthogonal operators are orthogonal matrices, i.e., they obey $O^T O = O O^T = I$ and thus $O^T = O^{-1}$. Note that this means that orthogonal matrices are [[Non-singular Matrix|non-singular]]. They are real matrices. In terms of components: $\sum_{\beta} o_{\alpha \gamma} o_{\beta \gamma} = \delta_{\alpha \beta},$ where $\delta_{\alpha \gamma}$ is the [[Kronecker Delta]]. ## Properties It is easy to show that $\det(O) =\pm 1$. Note that the orthogonal matrices of dimension $n\times n$ form the [[Orthogonal Group|orthogonal group]] $O(n)$.