Unitary matrix

Unitary matrix

From Wikipedia, the free encyclopedia
In mathematics, a unitary matrix is an ntimes n complex matrix U satisfying the condition
U^{dagger} U = UU^{dagger} = I_n,
where In 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 (realinner 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:
  1. U , is unitary
  2. U^{dagger} , is unitary
  3. the columns of U , form an orthonormal basis of mathbb{C}^n with respect to this inner product
  4. the rows of U , form an orthonormal basis of mathbb{C}^n with respect to this inner product
  5. U , is an isometry with respect to the norm from this inner product
  6. U , is a normal matrix with eigenvalues lying on the unit circle.
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