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{{for|matrices with  orthogonality  over the [[real number]] field|orthogonal matrix}}
{{for|the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1|unitarity}}

In [[mathematics]], a [[complex number|complex]] [[Matrix (mathematics)#Square matrices|square]] [[matrix(mathematics)|matrix]] {{math|''U''}} is '''unitary''' if its [[conjugate transpose]] {{math|''U''<sup>∗</sup>}} is also its [[inverse matrix|inverse]]—that is, if

::<math>U^* U = UU^* = I ,</math>

where {{mvar|I}} is the [[identity matrix]]. 

In physics, especially in quantum mechanics, the [[Hermitian conjugate]] of a matrix is denoted by a [[Dagger|dagger]] ([[†]]) and the equation above becomes

::<math>U^\dagger U = UU^\dagger = I .</math>

The real analogue of a unitary matrix is an [[orthogonal matrix]]. Unitary matrices have significant importance in quantum mechanics because they preserve [[Norm (mathematics)|norms]], and thus, [[probability amplitude]]s.

==Properties==
For any unitary matrix {{mvar|U}} of finite size, the following hold:
* Given two complex vectors {{mvar|x}} and {{mvar|y}}, multiplication by {{mvar|U}} preserves their [[inner product]]; that is, {{math| ⟨''Ux'', ''Uy''⟩ {{=}} ⟨''x'', ''y''⟩}}.
* {{mvar|U}} is [[normal matrix|normal]].
* {{mvar|U}} is [[diagonalizable matrix|diagonalizable]]; that is, {{mvar|U}} is [[similar matrix|unitarily similar]] to a diagonal matrix, as a consequence of the [[spectral theorem]].  Thus, {{mvar|U}} has a decomposition of the form
:: <math>U = VDV^*,</math> 
: where {{mvar|V}} is unitary, and {{mvar|D}} is diagonal and unitary.
* <math>\left|\operatorname{det}(U)\right| = 1</math>.
* Its [[Eigenvector#Eigenspaces of a matrix|eigenspaces]] are orthogonal.
* {{mvar|U}} can be written as {{mvar|U}}&nbsp;= {{mvar|e}}<sup>{{mvar|i}}{{mvar|H}}</sup>, where {{mvar|e}} indicates [[matrix exponential]], {{mvar|i}} is the imaginary unit, and {{mvar|H}} is a [[Hermitian matrix]].

For any nonnegative [[integer]] ''n'', the set of all ''n'' × ''n'' unitary matrices with matrix multiplication forms a [[group (mathematics)|group]], called the [[unitary group]] U(''n'').

Any square matrix with unit Euclidean norm is the average of two unitary matrices.<ref>{{cite journal| first1=Chi-Kwong|last1= Li |first2= Edward|last2= Poon|doi=10.1080/03081080290025507|title=Additive decomposition of real matrices| year=2002| journal=Linear and Multilinear Algebra| volume=50| issue=4| pages=321–326}}</ref>

==Equivalent conditions==
If ''U'' is a square, complex matrix, then the following conditions are equivalent:
#''U'' is unitary.
#''U''<sup>∗</sup>  is unitary.
#''U'' is invertible with {{nowrap|1=''U''<sup>−1</sup> = ''U''<sup>∗</sup>}}.
# The columns of ''U'' form an [[orthonormal basis]] of <math>\mathbb{C}^n</math> with respect to the usual inner product.
# The rows of ''U'' form an orthonormal basis of <math>\mathbb{C}^n</math> with respect to the usual inner product.
# ''U'' is an [[isometry]] with respect to the usual norm.
# ''U'' is a [[normal matrix]] with [[eigenvalues]] lying on the [[unit circle]].

==Elementary constructions==

=== 2 × 2 unitary matrix ===
The general expression of a {{nowrap|2 × 2}} unitary matrix is:

:<math>U = 
\begin{bmatrix}
a & b \\
-e^{i\varphi} b^* & e^{i\varphi} a^* \\
\end{bmatrix},\qquad \left| a \right| ^2 + \left| b \right| ^2 = 1 , </math>

which depends on 4 real parameters (the phase of {{mvar|a}}, the phase of {{mvar|b}}, the relative magnitude between {{mvar|a}} and {{mvar|b}}, and the angle {{mvar|φ}}). The [[determinant]] of such a matrix is:
:<math>
\det(U)=e^{i \varphi} .
</math>

The sub-group of those elements <math>U</math> with <math>\det(U) = 1</math> is called the [[Special unitary group#n = 2|special unitary group]] SU(2).

The matrix {{mvar|U}} can also be written in this alternative form:
:<math>U = 
e^{i\varphi /2}\begin{bmatrix}
e^{i\varphi_1} \cos \theta & e^{i\varphi_2} \sin \theta \\
-e^{-i\varphi_2} \sin \theta & e^{-i\varphi_1} \cos \theta \\
\end{bmatrix} , </math>

which, by introducing {{nowrap|1=''φ''<sub>1</sub> = ''ψ'' + Δ}} and {{nowrap|1=''φ''<sub>2</sub> = ''ψ'' − Δ}}, takes the following factorization:

:<math>U = 
e^{i\varphi /2}\begin{bmatrix}
e^{i\psi} & 0 \\
0 & e^{-i\psi}
\end{bmatrix}
\begin{bmatrix}
\cos \theta  & \sin \theta \\
-\sin \theta & \cos \theta \\
\end{bmatrix} 
\begin{bmatrix}
e^{i\Delta} & 0 \\
0 & e^{-i\Delta}
\end{bmatrix} .
</math>

This expression highlights the relation between {{nowrap|2 × 2}} unitary matrices and {{nowrap|2 × 2}} [[Orthogonal matrix|orthogonal matrices]] of angle {{mvar|θ}}.

Another factorization introduced in <ref>{{Cite journal|last=Führ|first=Hartmut|last2=Rzeszotnik|first2=Ziemowit|year=2018|title=A note on factoring unitary matrices|url=https://doi.org/10.1016/j.laa.2018.02.017|journal=Linear Algebra and its Applications|volume=547|pages=32–44|doi=10.1016/j.laa.2018.02.017|issn=0024-3795}}</ref>:

<math>U = 
\begin{bmatrix}
\cos \alpha  & -\sin \alpha \\
\sin \alpha & \cos \alpha \\
\end{bmatrix} 
\begin{bmatrix}
e^{i\xi} & 0 \\
0 & e^{i\zeta}
\end{bmatrix}
\begin{bmatrix}
\cos \beta  & \sin \beta \\
-\sin \beta & \cos \beta \\
\end{bmatrix} .
</math>

Many other factorizations of a unitary matrix in basic matrices are possible.

==See also==
* [[Orthogonal matrix]]
* [[Hermitian matrix]]
* [[Symplectic matrix]]
* [[Orthogonal group | Orthogonal group O(''n'')]]
* [[Orthogonal group | Special orthogonal group SO(''n'')]]
* [[Unitary group | Unitary group U(''n'')]]
* [[Special unitary group | Special Unitary group SU(''n'')]]
* [[Unitary operator]]
* [[Matrix decomposition]]
* [[Quantum gate]]

== References ==
{{reflist}}

== External links ==
* {{MathWorld|urlname=UnitaryMatrix |title=Unitary Matrix |last=Rowland|first= Todd}}
* {{SpringerEOM|id=U/u095540|title=Unitary matrix |first=O. A. |last=Ivanova}}
* {{cite web |title=Show that the eigenvalues of a unitary matrix have modulus 1 |work=[[Stack Exchange]] |date=March 28, 2016 |url=https://math.stackexchange.com/q/1717713 }}

{{Template:Matrix classes}}

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[[Category:Matrices]]
[[Category:Unitary operators]]