# Unitary matrix

In mathematics, a complex square matrix *U* is **unitary** if

where *I* is the identity matrix and *U** is the conjugate transpose of *U*. 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.

## 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,

*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

- .
- Its eigenspaces are orthogonal.
- For any nonnegative 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]} *U*can be written as*U*=e^{Template:MvarH}where e indicates matrix exponential,**Template:Mvar**is the imaginary unit and*H*is an Hermitian matrix.

## 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

### 2x2 Unitary matrix

The general expression of a 2x2 unitary matrix is:

which depends on 4 real parameters. The determinant of such a matrix is:

If *φ=0*, the group created by *U* is called special unitary group *SU(2)*.

Matrix *U* can also be written in this alternative form:

which, by introducing *φ _{1} = ψ + Δ* and

*φ*, takes the following factorization:

_{2}= ψ - ΔThis expression highlights the relation between 2x2 unitary matrices and 2x2 orthogonal matrices of angle *θ*.

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

### 3x3 Unitary matrix

The general expression of 3x3 unitary matrix is:^{[2]}

where *φ _{n}, n=1,...,5* are arbitrary real numbers, while

*K*is the Cabibbo–Kobayashi–Maskawa matrix.

## See also

- Orthogonal matrix
- Hermitian matrix
- Symplectic matrix
- Unitary group
- Special unitary group
- Unitary operator
- Matrix decomposition
- Identity matrix
- Quantum gate

## References

- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Template:Cite arXiv

## External links

- Weisstein, Eric W., "Unitary Matrix",
*MathWorld*. - {{#invoke:citation/CS1|citation

|CitationClass=citation }}