Unitary matrix

In mathematics, a complex square matrix U is unitary if its conjugate transpose U^∗ is also its inverse – that is, if

U^* U = UU^* = I ,

where I is the identity matrix. In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger (†) and the equation above becomes

U^\dagger U = UU^\dagger = I .

The real 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 U of finite size, the following hold:

Given two complex vectors x and y, multiplication by U preserves their inner product; that is,

\langle Ux, Uy \rangle = \langle x, y \rangle

U is normal
U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus U has a decomposition of the form

U = VDV^*\;

where V is unitary and D is diagonal and unitary.

$\left| \det ( U ) \right| = 1$ .
Its eigenspaces are orthogonal.
U can be written as U = e^{$i$ H}, where e indicates matrix exponential, $i$ is the imaginary unit and H is a Hermitian matrix.

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

Any square matrix with unit Euclidean norm is the average of two unitary matrices.^[1]

Equivalent conditions

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

U is unitary.
U^∗ is unitary.
U is invertible with U⁻¹ = U^∗.
The columns of U form an orthonormal basis of $\mathbb{C}^n$ with respect to the usual inner product.
The rows of U form an orthonormal basis of $\mathbb{C}^n$ 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 2 × 2 unitary matrix is:

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

which depends on 4 real parameters. The determinant of such a matrix is:

\det(U)=e^{i2\varphi} .

The sub-group of such elements in U where φ = 0 is called the special unitary group SU(2).

The matrix U can also be written in this alternative form:

U = e^{i\varphi}\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} ,

which, by introducing φ₁ = ψ + Δ and φ₂ = ψ − Δ, takes the following factorization:

U = e^{i\varphi}\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 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.

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

References

↑ Li, Chi-Kwong; Poon, Edward (2002). "Additive Decomposition of Real Matrices". Linear and Multilinear Algebra 50 (4): 321–326. doi:10.1080/03081080290025507.

External links

Weisstein, Eric W., "Unitary Matrix", MathWorld.
Ivanova, O. A. (2001), "Unitary matrix", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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