An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse:
which entails
where I is the identity matrix.
An orthogonal matrix A is necessarily invertible (with inverse A−1 = AT), unitary (A−1 = A*), and normal (A*A = AA*). The determinant of any orthogonal matrix is either +1 or −1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant +1 is a pure rotation, while every orthogonal matrix with determinant -1 is either a pure reflection, or a composition of reflection and rotation.
The complex analogue of an orthogonal matrix is a unitary matrix.
Read more about this topic: Matrix (mathematics), Square Matrices, Main Types
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