Orthogonal matrix

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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 $${\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}$$ 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: $${\displaystyle Q^{\mathrm {T} }=Q^{-1},}$$ 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.