Unitary matrix

In linear algebra, an invertible complex square matrix $U$ is unitary if its matrix inverse $U^{−1}$ equals its conjugate transpose $U^{*}$, that is, if

$$U^* U = UU^* = I,$$

where $I$ 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 (†), so the equation above is written

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

A complex matrix $U$ 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 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, $⟨Ux, Uy⟩ = ⟨x, y⟩$.
 * $U$ is normal ($$U^* U = UU^*$$).
 * $U$ 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.
 * $$\left|\det(U)\right| = 1$$. That is, $$\det(U)$$ will be on the unit circle of the complex plane.
 * Its eigenspaces are orthogonal.
 * $U$ can be written as $U = e^{iH}$, where $e$ indicates the matrix exponential, $i$ is the imaginary unit, and $H$ 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 $$\Complex^n$$ with respect to the usual inner product. In other words, $$U^*U = I$$.
 * 5) The rows of $$U$$ form an orthonormal basis of $$\Complex^n$$ with respect to the usual inner product. In other words, $$UU^* = I$$.
 * 6) $$U$$ is an isometry with respect to the usual norm. That is, $$\|Ux\|_2 = \|x\|_2$$ for all $$x \in \Complex^n$$, where $\|x\|_2 = \sqrt{\sum_{i=1}^n |x_i|^2}$.
 * 7) $$U$$ is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of $$U$$) with eigenvalues lying on the unit circle.

2 × 2 unitary matrix
One general expression of a 2 × 2 unitary matrix is

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

which depends on 4 real parameters (the phase of $a$, the phase of $b$, the relative magnitude between $a$ and $b$, and the angle $φ$). The form is configured so the determinant of such a matrix is $$ \det(U) = e^{i \varphi} ~. $$

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 $U$ can be written in this form: $$\ U = e^{i\varphi / 2} \begin{bmatrix} e^{i\alpha} \cos \theta & e^{i\beta} \sin \theta \\ -e^{-i\beta} \sin \theta & e^{-i\alpha} \cos \theta \\ \end{bmatrix}\ ,$$

where $$\ e^{i\alpha} \cos \theta = a\ $$ and $$\ e^{i\beta} \sin \theta = b\ ,$$ above, and the angles $$\ \varphi, \alpha, \beta, \theta\ $$ can take any values.

By introducing $$\ \alpha = \psi + \delta\ $$ and $$\ \beta = \psi - \delta\ ,$$ has the following factorization:

$$ U = e^{i\varphi /2} \begin{bmatrix} e^{i\psi} & 0 \\ 0 & e^{-i\psi} \end{bmatrix} \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{bmatrix} \begin{bmatrix} e^{i\delta} & 0 \\ 0 & e^{-i\delta} \end{bmatrix} ~. $$

This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle $θ$.

Another factorization is

$$U = \begin{bmatrix} \cos \rho &   -\sin \rho \\ \sin \rho &   \;\cos \rho \\ \end{bmatrix} \begin{bmatrix} e^{i\xi} & 0 \\ 0 & e^{i\zeta} \end{bmatrix} \begin{bmatrix} \;\cos \sigma &   \sin \sigma \\ -\sin \sigma  &   \cos \sigma \\ \end{bmatrix} ~. $$

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