Unitary Matrix

In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if

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

where 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

$U^\dagger U = UU^\dagger = I.$

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

Properties

For any unitary matrix of finite size, the following hold:
* Given two complex vectors and , multiplication by preserves their inner product; that is, .
* is normal ($U^* U = UU^*$).
* is diagonalizable; that is, is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, has a decomposition of the form $U = VDV^*,$ where is unitary, and is diagonal and unitary.
* $\left, \det(U)\ = 1$.
* Its eigenspaces are orthogonal.
* can be written as , where indicates the matrix exponential, is the imaginary unit, and is a Hermitian matrix.

For any nonnegative integer , the set of all unitary matrices with matrix multiplication forms a group, called the unitary group .

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:
# $U$ is unitary.
# $U^*$ is unitary.
# $U$ is invertible with $U^ = U^*$.
# The columns of $U$ form an orthonormal basis of $\Complex^n$ with respect to the usual inner product. In other words, $U^*U = I$.
# The rows of $U$ form an orthonormal basis of $\Complex^n$ with respect to the usual inner product. In other words, $UU^* = I$.
# $U$ is an isometry with respect to the usual norm. That is, $\, Ux\, _2 = \, x\, _2$ for all $x \in \Complex^n$, where $\, x\, _2 = \sqrt$.
# $U$ is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of $U$) with eigenvalues lying on the unit circle.

Elementary constructions

2 × 2 unitary matrix

The general expression of a unitary matrix is

$U = \begin a & b \\ -e^ b^* & e^ a^* \\ \end, \qquad \left, a \^2 + \left, b \^2 = 1,$

which depends on 4 real parameters (the phase of , the phase of , the relative magnitude between and , and the angle ). The determinant of such a matrix is
$\det(U) = e^.$

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

The matrix can also be written in this alternative form:
$U = e^ \begin e^ \cos \theta & e^ \sin \theta \\ -e^ \sin \theta & e^ \cos \theta \\ \end,$

which, by introducing and , takes the following factorization:

$U = e^\begin e^ & 0 \\ 0 & e^ \end \begin \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end \begin e^ & 0 \\ 0 & e^ \end.$

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

Another factorization is

$U = \begin \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \\ \end \begin e^ & 0 \\ 0 & e^ \end \begin \cos \beta & \sin \beta \\ -\sin \beta & \cos \beta \\ \end.$

Many other factorizations of a unitary matrix in basic matrices are possible., page 8See als
"Forbidden by symmetry"
/ref>

See also

* Hermitian matrix and Skew-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

External links

*
*
*

{{DEFAULTSORT:Unitary Matrix
Matrices
Unitary operators