In linear algebra, 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 adjoint 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.

## 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, ⟨*U***x**,*U***y**⟩ = ⟨**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 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.^{[1]}

## Equivalent conditions

If *U* is a square, complex matrix, then the following conditions are equivalent:^{[2]}

- is unitary.
- is unitary.
- is invertible with .
- The columns of form an orthonormal basis of with respect to the usual inner product. In other words, .
- The rows of form an orthonormal basis of with respect to the usual inner product. In other words, .
- is an isometry with respect to the usual norm. That is, for all , where .
- is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of ) 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 is^{[3]}

Many other factorizations of a unitary matrix in basic matrices are possible.^{[4]}^{[5]}^{[6]}

## See also

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

## References

**^**Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices".*Linear and Multilinear Algebra*.**50**(4): 321–326. doi:10.1080/03081080290025507.**^**Horn, Roger A.; Johnson, Charles R. (2013).*Matrix Analysis*. Cambridge University Press. doi:10.1017/9781139020411. ISBN 9781139020411.**^**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.**^**Williams, Colin P. (2011), Williams, Colin P. (ed.), "Quantum Gates",*Explorations in Quantum Computing*, Texts in Computer Science, London: Springer, p. 82, doi:10.1007/978-1-84628-887-6_2, ISBN 978-1-84628-887-6, retrieved 2021-05-14**^**Nielsen, Michael A.; Chuang, Isaac (2010).*Quantum Computation and Quantum Information*. Cambridge: Cambridge University Press. p. 20. ISBN 978-1-10700-217-3. OCLC 43641333.**^**Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald (1995-11-01). "Elementary gates for quantum computation".*Physical Review A*. American Physical Society (APS).**52**(5): 3457–3467. arXiv:quant-ph/9503016. doi:10.1103/physreva.52.3457. ISSN 1050-2947., page 8

## External links

- Weisstein, Eric W. "Unitary Matrix".
*MathWorld*. Todd Rowland. - Ivanova, O. A. (2001) [1994], "Unitary matrix",
*Encyclopedia of Mathematics*, EMS Press - "Show that the eigenvalues of a unitary matrix have modulus 1".
*Stack Exchange*. March 28, 2016.