Unitary matrix

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In mathematics, a complex square matrix U is unitary if

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

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because 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,
where V is unitary and D is diagonal and unitary.

Equivalent conditions

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 with respect to the usual inner product.
  5. The rows of U form an orthonormal basis of 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.

Elementary constructions

2x2 Unitary matrix

The general expression of a 2x2 unitary matrix is:

which depends on 4 real parameters. The determinant of such a matrix is:

If φ=0, the group created by U is called special unitary group SU(2).

Matrix U can also be written in this alternative form:

which, by introducing φ1 = ψ + Δ and φ2 = ψ - Δ, takes the following factorization:

This expression highlights the relation between 2x2 unitary matrices and 2x2 orthogonal matrices of angle θ.

Many other factorizations of a unitary matrix in basic matrices are possible.

3x3 Unitary matrix

The general expression of 3x3 unitary matrix is:[2]

where φn, n=1,...,5 are arbitrary real numbers, while K is the Cabibbo–Kobayashi–Maskawa matrix.

See also


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External links

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