# Unitary matrix

In mathematics, a complex square matrix U is unitary if

${\displaystyle U^{*}U=UU^{*}=I\,}$

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

${\displaystyle U^{\dagger }U=UU^{\dagger }=I.\,}$

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, the following hold:

• Given two complex vectors x and y, multiplication by U preserves their inner product; that is,
${\displaystyle \langle Ux,Uy\rangle =\langle x,y\rangle }$.
${\displaystyle U=VDV^{*}\;}$
where V is unitary and D is diagonal and unitary.

## 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 ${\displaystyle \mathbb {C} ^{n}}$ with respect to the usual inner product.
5. The rows of U form an orthonormal basis of ${\displaystyle \mathbb {C} ^{n}}$ 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

### 2x2 Unitary matrix

The general expression of a 2x2 unitary matrix is:

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

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

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

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:

${\displaystyle U=e^{i\varphi }{\begin{bmatrix}\cos \theta e^{i\varphi _{1}}&\sin \theta e^{i\varphi _{2}}\\-\sin \theta e^{-i\varphi _{2}}&\cos \theta e^{-i\varphi _{1}}\\\end{bmatrix}},}$

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

${\displaystyle U=e^{i\varphi }{\begin{bmatrix}e^{i\psi }&0\\0&e^{-i\psi }\end{bmatrix}}{\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}e^{i\Delta }&0\\0&e^{-i\Delta }\end{bmatrix}}.}$

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.

### 3x3 Unitary matrix

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

${\displaystyle U={\begin{bmatrix}1&0&0\\0&e^{j\varphi _{4}}&0\\0&0&e^{j\varphi _{5}}\end{bmatrix}}K{\begin{bmatrix}e^{j\varphi _{1}}&0&0\\0&e^{j\varphi _{2}}&0\\0&0&e^{j\varphi _{3}}\end{bmatrix}}}$

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