Unitary matrix

For any unitary matrix {{mvar|U}} of finite size, the following hold:

• Given two complex vectors {{math|x}} and {{math|y}}, multiplication by {{mvar|U}} preserves their inner product; that is, {{math|1=⟨Ux, Uy⟩ = ⟨x, y⟩}}.
• {{mvar|U}} is normal ($U^*\; U\; =\; UU^*$).
• {{mvar|U}} is diagonalizable; that is, {{mvar|U}} is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, {{mvar|U}} has a decomposition of the form $U\; =\; VDV^*,$ where {{mvar|V}} is unitary, and {{mvar|D}} is diagonal and unitary.
• The eigenvalues of $U$ lie on the unit circle, as does $\backslash det(U)$.
• The eigenspaces of $U$ are orthogonal.
• {{mvar|U}} can be written as {{math|1=U = e^iH}}, where {{mvar|e}} indicates the matrix exponential, {{mvar|i}} is the imaginary unit, and {{mvar|H}} is a Hermitian matrix.

For any nonnegative integer {{math|n}}, the set of all {{math|n × n}} unitary matrices with matrix multiplication forms a group, called the unitary group {{math|U(n)}}.

Every square matrix with unit Euclidean norm is the average of two unitary matrices.{{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|s2cid= 120125694 }}

If U is a square, complex matrix, then the following conditions are equivalent:{{Cite book | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher=Cambridge University Press | isbn=9781139020411 | year=2013 |doi=10.1017/CBO9781139020411 }}

1. $U$ is unitary.
2. $U^*$ is unitary.
3. $U$ is invertible with $U^\{-1\}\; =\; U^*$.
4. The columns of $U$ form an orthonormal basis of $\backslash Complex^n$ with respect to the usual inner product. In other words, $U^*U\; =\; I$.
5. The rows of $U$ form an orthonormal basis of $\backslash Complex^n$ with respect to the usual inner product. In other words, $UU^*\; =\; I$.
6. $U$ is an isometry with respect to the usual norm. That is, $\backslash |Ux\backslash |\_2\; =\; \backslash |x\backslash |\_2$ for all $x\; \backslash in\; \backslash Complex^n$, where $\backslash |x\backslash |\_2\; =\; \backslash sqrt\{\backslash sum\_\{i=1\}^n\; |x\_i|^2\}$.
7. $U$ is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of $U$) with eigenvalues lying on the unit circle.

One general expression of a {{nobr|2 × 2}} unitary matrix is

$$U\; =\; \backslash begin\{bmatrix\}$$

a & b \\

-e^{i\varphi} b^* & e^{i\varphi} a^* \\

\end{bmatrix},

\qquad

\left| a \right|^2 + \left| b \right|^2 = 1\ ,

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 form is configured so the determinant of such a matrix is

$$\backslash det(U)\; =\; e^\{i\; \backslash varphi\}\; ~.$$

The sub-group of those elements $\backslash \; U\backslash$ with $\backslash \; \backslash det(U)\; =\; 1\backslash$ is called the special unitary group SU(2).

Among several alternative forms, the matrix {{mvar|U}} can be written in this form:

$$\backslash \; U\; =\; e^\{i\backslash varphi\; /\; 2\}\; \backslash begin\{bmatrix\}$$

e^{i\alpha} \cos \theta & e^{i\beta} \sin \theta \\

-e^{-i\beta} \sin \theta & e^{-i\alpha} \cos \theta \\

\end{bmatrix}\ ,

where $\backslash \; e^\{i\backslash alpha\}\; \backslash cos\; \backslash theta\; =\; a\backslash$ and $\backslash \; e^\{i\backslash beta\}\; \backslash sin\; \backslash theta\; =\; b\backslash \; ,$ above, and the angles $\backslash \; \backslash varphi,\; \backslash alpha,\; \backslash beta,\; \backslash theta\backslash$ can take any values.

By introducing $\backslash \; \backslash alpha\; =\; \backslash psi\; +\; \backslash delta\backslash$ and $\backslash \; \backslash beta\; =\; \backslash psi\; -\; \backslash delta\backslash \; ,$ has the following factorization:

$$U\; =\; e^\{i\backslash varphi\; /2\}\; \backslash begin\{bmatrix\}$$

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} ~.

This expression highlights the relation between {{nobr|2 × 2}} unitary matrices and {{nobr|2 × 2}} orthogonal matrices of angle {{mvar|θ}}.

Another factorization is{{cite journal |last1=Führ |first1=Hartmut |last2=Rzeszotnik |first2=Ziemowit |year=2018 |title=A note on factoring unitary matrices |journal=Linear Algebra and Its Applications |volume=547 |pages=32–44 |doi=10.1016/j.laa.2018.02.017 |s2cid=125455174 |issn=0024-3795|doi-access=free }}

$$U\; =\; \backslash begin\{bmatrix\}$$

\cos \rho & -\sin \rho \\

\sin \rho & \;\cos \rho \\

\end{bmatrix}

\begin{bmatrix}

e^{i\xi} & 0 \\

0 & e^{i\zeta}

\end{bmatrix}

\begin{bmatrix}

\;\cos \sigma & \sin \sigma \\

-\sin \sigma & \cos \sigma \\

\end{bmatrix} ~.

Many other factorizations of a unitary matrix in basic matrices are possible.{{cite book |last=Williams |first=Colin P. |year=2011 |section=Quantum gates |title=Explorations in Quantum Computing |pages=82 |editor-last=Williams |editor-first=Colin P. |series=Texts in Computer Science |place=London, UK |publisher=Springer |lang=en |doi=10.1007/978-1-84628-887-6_2 |isbn=978-1-84628-887-6}}{{cite book |last1=Nielsen |first1=M.A. |author1-link=Michael Nielsen |last2=Chuang |first2=Isaac |author2-link=Isaac Chuang |year=2010 |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |isbn=978-1-10700-217-3 |place=Cambridge, UK |oclc=43641333 |url=https://www.cambridge.org/9781107002173 |page=20}}{{cite journal | last1=Barenco | first1=Adriano | last2=Bennett | first2=Charles H. | last3=Cleve | first3=Richard | last4=DiVincenzo | first4=David P. | last5=Margolus | first5=Norman | last6=Shor | first6=Peter | last7=Sleator | first7=Tycho | last8=Smolin | first8=John A. | last9=Weinfurter | first9=Harald | display-authors=6 | date=1995-11-01 | df=dmy-all | title=Elementary gates for quantum computation | journal=Physical Review A | publisher=American Physical Society (APS) | volume=52 | issue=5 | issn=1050-2947 | doi=10.1103/physreva.52.3457 | pages=3457–3467, esp.p. 3465 | pmid=9912645 | arxiv=quant-ph/9503016 | bibcode=1995PhRvA..52.3457B | s2cid=8764584 }}{{cite journal |last=Marvian |first=Iman |date=2022-01-10 |df=dmy-all |title=Restrictions on realizable unitary operations imposed by symmetry and locality |journal=Nature Physics |volume=18 |issue=3 |pages=283–289 |arxiv=2003.05524 |doi=10.1038/s41567-021-01464-0 |bibcode=2022NatPh..18..283M |s2cid=245840243 |issn=1745-2481 |lang=en |url=https://www.nature.com/articles/s41567-021-01464-0}}{{cite journal |last=Jarlskog |first = Cecilia |date=2006 |title=Recursive parameterisation and invariant phases of unitary matrices |journal = Journal of Mathematical Physics |volume = 47 |issue = 1 |page = 013507 |doi = 10.1063/1.2159069 |arxiv=math-ph/0510034|bibcode = 2006JMP....47a3507J }}{{cite journal |author=Alhambra, Álvaro M. |date=10 January 2022 |title=Forbidden by symmetry |journal=Nature Physics |volume=18 |issue=3 |pages=235–236 |issn=1745-2481 |doi=10.1038/s41567-021-01483-x |bibcode=2022NatPh..18..235A |s2cid=256745894 |department=News & Views |url=https://www.nature.com/articles/s41567-021-01483-x.epdf?sharing_token=cb9JltmO0c_GuA_zyl_Hn9RgN0jAjWel9jnR3ZoTv0N2eMl-wQgGXVDdGkt0dHblV7Y2XiScmBn7eBbLkk2wN8fTlUuAcjP8wOfRS37lCMALVlmwQ72SNethITLikGw1OaeWVi_dwhQkvNW-wS5wsbz_fc5pIxAQO3XEghzc25Y%3D |quote=The physics of large systems is often understood as the outcome of the local operations among its components. Now, it is shown that this picture may be incomplete in quantum systems whose interactions are constrained by symmetries.}}

{{div col begin |colwidth=15em}}

• Hermitian matrix
• Skew-Hermitian matrix
• Matrix decomposition
• Orthogonal group O(n)
• Special orthogonal group SO(n)
• Orthogonal matrix
• Semi-orthogonal matrix
• Quantum logic gate
• Special Unitary group SU(n)
• Symplectic matrix
• Unitary group U(n)
• Unitary operator

{{div col end}}

{{reflist|25em}}

• {{MathWorld|urlname=UnitaryMatrix |title=Unitary Matrix |others=Todd Rowland}}
• {{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 }}

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Category:Matrices (mathematics)

Category:Unitary operators