Unitary matrix

In linear algebra, an invertible complex square matrix Template:Mvar is unitary if its matrix inverse U⁻¹ equals its conjugate transpose U^*, 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 1.

For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve the normalization of state vectors and the inner products between them.

Properties

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

• Given two complex vectors x and y, multiplication by Template:Mvar preserves their inner product; that is, ⟨Ux, Uy⟩ = ⟨x, y⟩.
• 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 U = e^iH, where Template:Mvar indicates the matrix exponential, Template:Mvar is the imaginary unit, and Template:Mvar is a Hermitian matrix.

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

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:Times 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:Times unitary matrices and Template:Times 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