Unitary matrix

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In mathematics, a unitary matrix is an $ntimes n$ complex matrix U satisfying the condition

$U^{dagger} U = UU^{dagger} = I_n,$

where I_n is the identity matrix in n dimensions and $U^{dagger}$ 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 $U^{dagger} ,$

$U^{-1} = U^{dagger} ,;$

A unitary matrix in which all entries are real is an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner productof two real vectors,

$langle Gx, Gy rangle = langle x, y rangle$

so also a unitary matrix U satisfies

$langle Ux, Uy rangle = langle x, y rangle$

for all complex vectors x and y, where $langlecdot,cdotrangle$ stands now for the standard inner product on $mathbb{C}^n$ .

If $U ,$ is an n by n matrix then the following are all equivalent conditions:

$U ,$ is unitary
$U^{dagger} ,$ is unitary
the columns of $U ,$ form an orthonormal basis of $mathbb{C}^n$ with respect to this inner product
the rows of $U ,$ form an orthonormal basis of $mathbb{C}^n$ with respect to this inner product
$U ,$ is an isometry with respect to the norm from this inner product
$U ,$ is a normal matrix with eigenvalues lying on the unit circle.

http://en.wikipedia.org/wiki/Unitary_matrix

2 thoughts on “Unitary matrix”

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