Unitary matrix

In linear algebra, an invertible complex square matrix Template:Mvar is unitary if its matrix inverse Template:Math equals its conjugate transpose Template:Math, that is, if

<math display=block>U^* U = UU^* = I,</math>

where Template:Mvar is the identity matrix.

In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (Template:Tmath), so the equation above is written

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

A complex matrix Template:Mvar is special unitary if it is unitary and its matrix determinant equals Template:Math.

For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

Properties

For any unitary matrix Template:Mvar of finite size, the following hold:

• Given two complex vectors Template:Math and Template:Math, multiplication by Template:Mvar preserves their inner product; that is, Template:Math.
• Template:Mvar is normal (<math>U^* U = UU^*</math>).
• Template:Mvar is diagonalizable; that is, Template:Mvar is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, Template:Mvar has a decomposition of the form <math>U = VDV^*,</math> where Template:Mvar is unitary, and Template:Mvar is diagonal and unitary.
• The eigenvalues of <math>U</math> lie on the unit circle, as does <math>\det(U)</math>.
• The eigenspaces of <math>U</math> are orthogonal.
• Template:Mvar can be written as Template:Math, where Template:Mvar indicates the matrix exponential, Template:Mvar is the imaginary unit, and Template:Mvar is a Hermitian matrix.

For any nonnegative integer Template:Math, the set of all Template:Math unitary matrices with matrix multiplication forms a group, called the unitary group Template:Math.

Every square matrix with unit Euclidean norm is the average of two unitary matrices.<ref>Template:Cite journal</ref>

Equivalent conditions

If U is a square, complex matrix, then the following conditions are equivalent:<ref>Template:Cite book</ref>

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

Elementary constructions

2 × 2 unitary matrix

One general expression of a Template:Nobr unitary matrix is

<math display=block>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 Template:Mvar, the phase of Template:Mvar, the relative magnitude between Template:Mvar and Template:Mvar, and the angle Template:Mvar). The form is configured so the determinant of such a matrix is <math display=block> \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 SU(2).

Among several alternative forms, the matrix Template:Mvar can be written in this form: <math display=block>\ U = e^{i\varphi / 2} \begin{bmatrix}

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

\end{bmatrix}\ ,</math>

where <math>\ e^{i\alpha} \cos \theta = a\ </math> and <math>\ e^{i\beta} \sin \theta = b\ ,</math> above, and the angles <math>\ \varphi, \alpha, \beta, \theta\ </math> can take any values.

By introducing <math>\ \alpha = \psi + \delta\ </math> and <math>\ \beta = \psi - \delta\ ,</math> has the following factorization:

<math display=block> 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 Template:Nobr unitary matrices and Template:Nobr orthogonal matrices of angle Template:Mvar.

Another factorization is<ref>Template:Cite journal</ref>

<math display=block>U = \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} ~. </math>

Many other factorizations of a unitary matrix in basic matrices are possible.<ref>Template:Cite book</ref><ref>Template:Cite book</ref><ref name=Barenco>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

See also

• 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

External links

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:UnitaryMatrix%7CUnitaryMatrix.html}} |title = Unitary Matrix |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• Template:SpringerEOM
• {{#invoke:citation/CS1|citation

|CitationClass=web }}