# Unitary matrix

Short description: Complex matrix whose conjugate transpose equals its inverse

In linear algebra, an invertible complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if

$\displaystyle{ U^* U = UU^* = UU^{-1} = I, }$

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

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

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 ($\displaystyle{ U^* U = UU^* }$).
• 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 $\displaystyle{ U = VDV^*, }$ where V is unitary, and D is diagonal and unitary.
• $\displaystyle{ \left|\det(U)\right| = 1 }$.
• Its eigenspaces are orthogonal.
• U can be written as U = eiH, 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).

Any 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. $\displaystyle{ U }$ is unitary.
2. $\displaystyle{ U^* }$ is unitary.
3. $\displaystyle{ U }$ is invertible with $\displaystyle{ U^{-1} = U^* }$.
4. The columns of $\displaystyle{ U }$ form an orthonormal basis of $\displaystyle{ \Complex^n }$ with respect to the usual inner product. In other words, $\displaystyle{ U^*U = I }$.
5. The rows of $\displaystyle{ U }$ form an orthonormal basis of $\displaystyle{ \Complex^n }$ with respect to the usual inner product. In other words, $\displaystyle{ UU^* = I }$.
6. $\displaystyle{ U }$ is an isometry with respect to the usual norm. That is, $\displaystyle{ \|Ux\|_2 = \|x\|_2 }$ for all $\displaystyle{ x \in \Complex^n }$, where $\displaystyle{ \|x\|_2 = \sqrt{\sum_{i=1}^n |x_i|^2} }$.
7. $\displaystyle{ U }$ is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of $\displaystyle{ U }$) with eigenvalues lying on the unit circle.

## Elementary constructions

### 2 × 2 unitary matrix

One general expression of a 2 × 2 unitary matrix is

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

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

The sub-group of those elements $\displaystyle{ \ U\ }$ with $\displaystyle{ \ \det(U) = 1\ }$ is called the special unitary group SU(2).

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

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

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

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

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

Another factorization is

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

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