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In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:
An orthogonal matrix Q is necessarily invertible , unitary , where Q is the Hermitian adjoint of Q, and therefore normal over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation.
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