Editing Orthogonal matrix (section)

{{Short description|Real square matrix whose columns and rows are orthogonal unit vectors}}
{{for|matrices with orthogonality over the [[complex number]] field|unitary matrix}}
{{More footnotes needed|date=May 2023}}
In [[linear algebra]], an '''orthogonal matrix''', or '''orthonormal matrix''', is a real [[square matrix]] whose columns and rows are [[Orthonormality|orthonormal]] [[Vector (mathematics and physics)|vectors]].

One way to express this is
<math display="block">Q^\mathrm{T} Q = Q Q^\mathrm{T} = I,</math>
where {{math|''Q''<sup>T</sup>}} is the [[transpose]] of {{mvar|Q}} and {{mvar|I}} is the [[identity matrix]].

This leads to the equivalent characterization: a matrix {{mvar|Q}} is orthogonal if its transpose is equal to its [[Invertible matrix|inverse]]:
<math display="block">Q^\mathrm{T}=Q^{-1},</math>
where {{math|''Q''<sup>−1</sup>}} is the inverse of {{mvar|Q}}.

An orthogonal matrix {{mvar|Q}} is necessarily invertible (with inverse {{math|1=''Q''<sup>−1</sup> = ''Q''<sup>T</sup>}}), [[Unitary matrix|unitary]] ({{math|1=''Q''<sup>−1</sup> = ''Q''<sup>∗</sup>}}), where {{math|1=''Q''<sup>∗</sup>}} is the [[Hermitian adjoint]] ([[conjugate transpose]]) of {{mvar|Q}}, and therefore [[Normal matrix|normal]] ({{math|1=''Q''<sup>∗</sup>''Q'' = ''QQ''<sup>∗</sup>}}) over the [[real number]]s. The [[determinant]] of any orthogonal matrix is either +1 or −1. As a [[Linear map|linear transformation]], an orthogonal matrix preserves the [[inner product]] of vectors, and therefore acts as an [[isometry]] of [[Euclidean space]], such as a [[Rotation (mathematics)|rotation]], [[Reflection (mathematics)|reflection]] or [[Improper rotation|rotoreflection]]. In other words, it is a [[unitary transformation]].

The set of {{math|''n'' × ''n''}} orthogonal matrices, under multiplication, forms the [[group (mathematics)|group]] {{math|O(''n'')}}, known as the [[orthogonal group]]. The [[subgroup]] {{math|SO(''n'')}} consisting of orthogonal matrices with determinant +1 is called the [[Orthogonal group#special orthogonal group|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.