An operator, $A$, is _orthogonal_ if and only if it preserves the [real inner product](Inner%20products.md#Real%20inner%20products). That is, if $(A^T \mathbf{x}, A\mathbf{y}) = (\mathbf{x}, \mathbf{y})$ where $\mathbf{x}$ and $\mathbf{y}$ are elements of a [real inner product space](Real%20vector%20spaces.md#real%20inner%20product%20spaces) and $A^T$ is the [transpose](Transpose%20of%20a%20linear%20map.md). This also means that orthogonal operations on real vectors preserve length. This length preserving quality means that orthogonal operators are [linear isometries](Linear%20isometry.md) on a real inner product space. # Orthogonal matrices An orthogonal matrix is a real $n \times n$ matrix consisting of column vectors and row vectors that form an [orthonormal bases](Orthonormal%20bases.md). Note how the properties of orthogonal matrices mirror properties of unitary matrices where here we take the [transpose](Transpose%20of%20a%20linear%20map.md#transpose%20of%20a%20matrix) instead of the _conjugate transpose_ to get the inverse matrix. Thus, when restricted to matrix elements in $\mathbb{R}$ orthogonality is equivalent to [unitarity](Unitary%20operators.md#Unitary%20matrices). A matrix $A$ is orthogonal if and only if $A^T = A^{-1}$ where $(A^T)_{jk} = A_{kj}$. When acting on [real vector spaces](Real%20vector%20spaces.md) orthogonal matrices preserve the [inner product](Inner%20products.md). ## Orthogonal matrix groups The group of $n \times n$ orthogonal matrices is [[O(n)]] or $\mbox{O}(n,\mathbb{C})$ where the group is over complex entries. ## Determinant properties of orthogonal matrices [Orthogonal matrices](Orthogonal%20operators.md#Orthogonal%20matrices) have the following determinant properties: 1) $\det{(A^T A)} = |\det(A)|^2 =\det(I) = 1$ 2) $\det{(A)} = \pm 1$ #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices