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

$$U^{*}U=U{U}^{*}=I,$$

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

$$U^{†}U=U{U}^{†}=I.$$

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 ($U^{*}U=U{U}^{*}$).
• 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 $U=VD{V}^{*},$ where Template:Mvar is unitary, and Template:Mvar is diagonal and unitary.
• The eigenvalues of $U$ lie on the unit circle, as does $\det (U)$.
• The eigenspaces of $U$ 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.

Equivalent conditions

If U is a square, complex matrix, then the following conditions are equivalent:

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

Elementary constructions

2 × 2 unitary matrix

One general expression of a Template:Math unitary matrix is $$U=\left[\begin{array}{cc}a& b\\ -e^{i\phi }{b}^{*}& e^{i\phi }{a}^{*}\\ \end{array}\right],{\left|a\right|}^2+{\left|b\right|}^2=1,$$

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) and * is the complex conjugate. The form is configured so the determinant of such a matrix is $$\det (U)=e^{i\phi }.$$

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

Among several alternative forms, the matrix Template:Mvar can be written in this form: $$U=e^{i\phi /2}\left[\begin{array}{cc}{e}^{i\alpha }\cos \theta & e^{i\beta }\sin \theta \\ -e^{-i\beta }\sin \theta & e^{-i\alpha }\cos \theta \\ \end{array}\right],$$

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

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

$$U=e^{i\phi /2}\left[\begin{array}{cc}{e}^{i\psi }& 0\\ 0& e^{-i\psi }\end{array}\right]\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{array}\right]\left[\begin{array}{cc}{e}^{i\delta }& 0\\ 0& e^{-i\delta }\end{array}\right].$$

This expression highlights the relation between Template:Math unitary matrices and Template:Math orthogonal matrices of angle Template:Mvar.

Another factorization is

$$U=\left[\begin{array}{cc}\cos \rho & -\sin \rho \\ \sin \rho & \cos \rho \\ \end{array}\right]\left[\begin{array}{cc}{e}^{i\xi }& 0\\ 0& e^{i\zeta }\end{array}\right]\left[\begin{array}{cc}\cos \sigma & \sin \sigma \\ -\sin \sigma & \cos \sigma \\ \end{array}\right].$$

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

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

• Template:Main other