# Orthogonal matrix

## Real square matrix whose columns and rows are orthogonal unit vectors / From Wikipedia, the free encyclopedia

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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.

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One way to express this is
$Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,$
where *Q*^{T} 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} = *Q*^{T}), 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 rotation, reflection or rotoreflection. In other words, it is a unitary transformation.

The set of *n* × *n* orthogonal matrices, under multiplication, forms the group O(*n*), known as the orthogonal group. The subgroup SO(*n*) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.