Unitary matrix

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{math, number, function}

In mathematics, a unitary matrix is an n\times n complex matrix U satisfying the condition

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} \,

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

so also a unitary matrix U satisfies

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:

Contents

Properties

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

For any unitary matrix U, the following hold:

  • U \, is invertible, with U^{-1}=U^{\dagger}.
  • U^{\dagger} is also unitary.
  • U \, preserves length ("isometry"): \|Ux\|_2=\|x\|_2.
  • U \, has complex eigenvalues of modulus 1. [1]
  • For any n, the set of all n by n unitary matrices with matrix multiplication forms a group, called U(n).
  • Any unit-norm matrix is the average of two unitary matrices. As a consequence, every n \times n matrix M is a linear combination of two unitary matrices.[2]

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