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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
==Overview==
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