Concept

# Orthogonal matrix

Summary
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 Q^\mathrm{T} Q = Q Q^\mathrm{T} = I, where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: Q^\mathrm{T}=Q^{-1}, where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) 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 rotatio
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