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^iH, 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:

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.
5. The rows of U form an orthonormal basis of $\mathbb{C}^n$ with respect to the usual inner product.
6. U is an isometry with respect to the usual norm.
7. 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:

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

which depends on 4 real parameters (the phase of $a$, the phase of $b$, the relative magnitude between $a$ and $b$, and the angle $\theta$). The determinant of such a matrix is:

$\det(U)=e^{i \theta} .$

The sub-group of such elements in U where $\det(U)=1$ 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.

See also

• Orthogonal matrix
• Hermitian matrix
• Symplectic matrix
• Orthogonal group O(n)
• Special Orthogonal group SO(n)
• Unitary group U(n)
• Special Unitary group SU(n)
• Unitary operator
• Matrix decomposition
• Identity matrix
• Quantum gate

References

1. 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