算盤: Unitary matrix

2011-02-01

From Wikipedia, the free encyclopedia

In mathematics, a unitary matrix is an $n\times n$ 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 $\langle\cdot,\cdot\rangle$ 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.

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