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{{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]].
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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==
[[File:Matrix multiplication transpose.svg|thumb|275px|Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA<sup>T</sup> will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of A.]]
An orthogonal matrix is the real specialization of a unitary matrix, and thus always a [[normal matrix]]. Although we consider only real matrices here, the definition can be used for matrices with entries from any [[field (mathematics)|field]]. However, orthogonal matrices arise naturally from [[dot product]]s, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product,<ref>[http://tutorial.math.lamar.edu/Classes/LinAlg/OrthogonalMatrix.aspx "Paul's online math notes"]{{Citation broken|date=January 2013|note=See talk page.}}, Paul Dawkins, [[Lamar University]], 2008. Theorem 3(c)</ref> so, for vectors {{math|'''u'''}} and {{math|'''v'''}} in an {{mvar|n}}-dimensional real [[Euclidean space]]▼
▲An orthogonal matrix is the real specialization of a unitary matrix, and thus always a [[normal matrix]]. Although we consider only real matrices here, the definition can be used for matrices with entries from any [[field (mathematics)|field]]. However, orthogonal matrices arise naturally from [[dot product]]s, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product,<ref>[http://tutorial.math.lamar.edu/Classes/LinAlg/OrthogonalMatrix.aspx "Paul's online math notes"]{{
<math display="block">{\mathbf u} \cdot {\mathbf v} = \left(Q {\mathbf u}\right) \cdot \left(Q {\mathbf v}\right) </math>
where {{mvar|Q}} is an orthogonal matrix. To see the inner product connection, consider a vector {{math|'''v'''}} in an {{mvar|n}}-dimensional real [[Euclidean space]]. Written with respect to an orthonormal basis, the squared length of {{math|'''v'''}} is {{math|'''v'''<sup>T</sup>'''v'''}}. If a linear transformation, in matrix form {{math|''Q'''''v'''}}, preserves vector lengths, then
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\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{bmatrix}</math>    (rotation about the origin)
*<math>
\begin{bmatrix}
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0 & 1\\
1 & 0
\end{bmatrix}.</math>
The identity is also a permutation matrix.
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In the case of a linear system which is underdetermined, or an otherwise non-[[invertible matrix]], singular value decomposition (SVD) is equally useful. With {{mvar|A}} factored as {{math|''U''Σ''V''<sup>T</sup>}}, a satisfactory solution uses the Moore-Penrose [[pseudoinverse]], {{math|''V''Σ<sup>+</sup>''U''<sup>T</sup>}}, where {{math|Σ<sup>+</sup>}} merely replaces each non-zero diagonal entry with its reciprocal. Set {{math|'''x'''}} to {{math|''V''Σ<sup>+</sup>''U''<sup>T</sup>'''b'''}}.
The case of a square invertible matrix also holds interest. Suppose, for example, that {{mvar|A}} is a {{nowrap|3 × 3}} rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so {{mvar|A}} has gradually lost its true orthogonality. A [[Gram–Schmidt process]] could [[orthogonalization|orthogonalize]] the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The [[polar decomposition]] factors a matrix into a pair, one of which is the unique ''closest'' orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any [[matrix norm]] invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "[[Newton's method]]" approach due to {{harvtxt|Higham|1986}} ([[#CITEREFHigham1990|1990]]), repeatedly averaging the matrix with its inverse transpose. {{harvtxt|Dubrulle|
For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps
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This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically:
<math display="block">Q_{n + 1} = 2 M \left(Q_n^{-1} M + M^\mathrm{T} Q_n\right)^{-1}</math>
where {{math|1=''Q''<sub>0</sub> = ''M''}}.
These iterations are stable provided the [[condition number]] of {{mvar|M}} is less than three.<ref>[http://www.maths.manchester.ac.uk/~nareports/narep91.pdf "Newton's Method for the Matrix Square Root"] {{Webarchive|url=https://web.archive.org/web/20110929131330/http://www.maths.manchester.ac.uk/~nareports/narep91.pdf |date=2011-09-29 }}, Nicholas J. Higham, Mathematics of Computation, Volume 46, Number 174, 1986.</ref>
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There is no standard terminology for these matrices. They are variously called "semi-orthogonal matrices", "orthonormal matrices", "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".
For the case {{math|''n'' ≤ ''m''}}, matrices with orthonormal columns may be referred to as [[k-frame|
==See also==
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