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.


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]


  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]