# Unitary matrix

In mathematics, a complex square matrix U is unitary if

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

where I is the identity matrix and U* is the conjugate transpose of U. In physics, especially in quantum mechanics, the conjugate transpose 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 due to the fact 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,
$\langle Ux, Uy \rangle = \langle x, y \rangle$.
$U = VDV^*\;$
where V is unitary and D is diagonal and unitary.
• $|\det(U)|=1$.
• Its eigenspaces are orthogonal.
• For any positive 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.