In mathematics, a complex square matrix *U* is **unitary** if its conjugate transpose *U*^{∗} is also its inverse—that is, if

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

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.

## Contents

## 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 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

- where V is unitary, and D is diagonal and unitary.

- .
- Its eigenspaces are orthogonal.
- U can be written as U = e
^{iH}, where e indicates 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.^{[1]}

## Equivalent conditions

If *U* is a square, complex matrix, then the following conditions are equivalent:

*U*is unitary.*U*^{∗}is unitary.*U*is invertible with*U*^{−1}=*U*^{∗}.- The columns of
*U*form an orthonormal basis of with respect to the usual inner product. - The rows of
*U*form an orthonormal basis of with respect to the usual inner product. *U*is an isometry with respect to the usual norm.*U*is a normal matrix with eigenvalues lying on the unit circle.

## Elementary constructions

### 2 × 2 unitary matrix

The general expression of a 2 × 2 unitary matrix is:

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 determinant of such a matrix is:

The sub-group of those elements with is called the special unitary group SU(2).

The matrix U can also be written in this alternative form:

which, by introducing *φ*_{1} = *ψ* + Δ and *φ*_{2} = *ψ* − Δ, takes the following factorization:

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

Another factorization introduced in ^{[2]}:

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

## See also

- Orthogonal matrix
- Hermitian matrix
- Symplectic matrix
- Orthogonal group O(
*n*) - Special orthogonal group SO(
*n*) - Unitary group U(
*n*) - Special Unitary group SU(
*n*) - Unitary operator
- Matrix decomposition
- Quantum gate

## References

**^**Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices".*Linear and Multilinear Algebra*.**50**(4): 321–326. doi:10.1080/03081080290025507.**^**Führ, Hartmut; Rzeszotnik, Ziemowit (2018). "A note on factoring unitary matrices".*Linear Algebra and its Applications*.**547**: 32–44. doi:10.1016/j.laa.2018.02.017. ISSN 0024-3795.

## External links

- Weisstein, Eric W. "Unitary Matrix".
*MathWorld*. - Ivanova, O. A. (2001) [1994], "Unitary matrix", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - “Show that the eigenvalues of a unitary matrix have modulus 1” on Mathematics Stack Exchange