# 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,
$\langle Ux, Uy \rangle = \langle x, y \rangle$.
$U = VDV^*\;$
where V is unitary and D is diagonal and unitary.
• $|\det(U)|=1$.
• Its eigenspaces are orthogonal.
• U can be written as U=eiH where e indicates matrix exponential, i is the imaginary unit and H is an 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 2x2 unitary matrix is:

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

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

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

If φ=0, the group created by U is called special unitary group SU(2).

Matrix U can also be written in this alternative form:

$U = e^{i\varphi}\begin{bmatrix} \cos \theta e^{i\varphi_1} & \sin \theta e^{i\varphi_2}\\ -\sin \theta e^{-i\varphi_2}& \cos \theta e^{-i\varphi_1}\\ \end{bmatrix} ,$

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 2x2 unitary matrices and 2x2 orthogonal matrices of angle θ.

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

### 3×3 unitary matrix

The general expression of 3x3 unitary matrix is:[2]

$U = \begin{bmatrix} 1 & 0 & 0 \\ 0 & e^{j\varphi_4} & 0 \\ 0 & 0 & e^{j\varphi_5} \end{bmatrix} K \begin{bmatrix} e^{j\varphi_1} & 0 & 0 \\ 0 & e^{j\varphi_2} & 0 \\ 0 & 0 & e^{j\varphi_3} \end{bmatrix}$

where φn, n=1,...,5 are arbitrary real numbers, while K is the Cabibbo–Kobayashi–Maskawa matrix.