Unitary matrix

In mathematics, a unitary matrix is an "n" by "n" complex matrix "U" satisfying the condition

:$U^*\; U\; =\; UU^*\; =\; I\_n,$ where $I\_n,$ is the identity matrix and $U^*\; ,$ is the conjugate transpose (also called the Hermitian adjoint) of "U". Note this condition says that a matrix "U" is unitary if and only if it has an inverse which is equal to its conjugate transpose $U^*\; ,$

:$U^\{-1\}\; =\; U^*\; ,;$

A unitary matrix in which all entries are real is the same thing as an orthogonal matrix. Just as an orthogonal matrix "G" preserves the (real) inner product of two real vectors,:$langle\; Gx,\; Gy\; angle\; =\; langle\; x,\; y\; angle$so also a unitary matrix "U" satisfies:$langle\; Ux,\; Uy\; angle\; =\; langle\; x,\; y\; angle$for all "complex" vectors "x" and "y", where $langlecdot,cdot\; angle$ stands now for the standard inner product on C^"n". If $U\; ,$ is an "n" by "n" matrix then the following are all equivalent conditions:

#$U\; ,$ is unitary
#$U^*\; ,$ is unitary
# the columns of $U\; ,$ form an orthonormal basis of C^"n" with respect to this inner product
# the rows of $U\; ,$ form an orthonormal basis of C^"n" with respect to this inner product
# $U\; ,$ is an isometry with respect to the norm from this inner product

It follows from the isometry property that all eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e., they lie on the unit circle centered at 0 in the complex plane). The same is true for the determinant.

All unitary matrices are normal, and the spectral theorem therefore applies to them. Thus every unitary matrix "U" has a decomposition of the form

:$U\; =\; VSigma\; V^*;$

where "V" is unitary, and $Sigma$ is diagonal and unitary.

For any "n", the set of all "n" by "n" unitary matrices with matrix multiplication form a group.

Properties of unitary matrices

* $U$ is invertible
* $U^\{-1\}=U^*$
* |det($U$)| = 1
* $U^*$ is unitary
* Unitary matrices preserve length $|Ux|\_2=|x|\_2$

ee also

*Hermitian matrix
*symplectic matrix
*unitary group
*special unitary group
*unitary operator
*defect of a unitary matrix
*matrix decomposition

External links

* [http://mathworld.wolfram.com/UnitaryMatrix.html Unitary Matrix from Mathworld ]