Unitary matrix

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In mathematics, a complex square matrix U is unitary if

U^* U = UU^* = I \,

where I is the identity matrix and U* is the conjugate transpose of U. In physics, especially in quantum mechanics, the hermitian conjugate of a matrix is denoted by a dagger (†) and the equation above becomes

U^\dagger U = UU^\dagger = I. \,

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics due to the fact they preserve norms, and thus, probability amplitudes.

Properties[edit]

For any unitary matrix U, the following hold:

  • Given two complex vectors x and y, multiplication by U preserves their inner product; that is,
\langle Ux, Uy \rangle = \langle x, y \rangle.
U = VDV^*\;
where V is unitary and D is diagonal and unitary.
  • |\det(U)|=1.
  • Its eigenspaces are orthogonal.
  • For any nonnegative integer n, the set of all n by n unitary matrices with matrix multiplication forms a group, called the unitary group U(n).
  • Any square matrix with unit Euclidean norm is the average of two unitary matrices.[1]

Equivalent conditions[edit]

If U is a square, complex matrix, then the following conditions are equivalent:

  1. U is unitary.
  2. U* is unitary.
  3. U is invertible with U –1=U*.
  4. The columns of U form an orthonormal basis of \mathbb{C}^n with respect to the usual inner product.
  5. The rows of U form an orthonormal basis of \mathbb{C}^n with respect to the usual inner product.
  6. U is an isometry with respect to the usual norm.
  7. U is a normal matrix with eigenvalues lying on the unit circle.

See also[edit]

References[edit]

  1. ^ Li, Chi-Kwong; Poon, Edward (2002). "Additive Decomposition of Real Matrices". Linear and Multilinear Algebra 50 (4): 321–326. doi:10.1080/03081080290025507. 

External links[edit]