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Unitary matrix

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In mathematics, a complex square matrix U is unitary if its conjugate transpose U∗ is also its inverse – that is, if

Contents

where I is the identity matrix. 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.

Properties

For any unitary matrix U of finite size, the following hold:

  • Given two complex vectors x and y, multiplication by U preserves their inner product; that is,
    • ⟨ U x , U y ⟩ = ⟨ x , y ⟩ .
  • U is normal
  • U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus U has a decomposition of the form
    • where V is unitary and D is diagonal and unitary.
  • | det ( U ) | = 1 .
  • Its eigenspaces are orthogonal.
  • U can be written as U = eiH, where e indicates matrix exponential, i is the imaginary unit and H is a Hermitian matrix.

    • 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.

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

    2 × 2 unitary matrix

    The general expression of a 2 × 2 unitary matrix is:

    U = [ a b − e i θ b ∗ e i θ a ∗ ] , | a | 2 + | b | 2 = 1 ,

    which depends on 4 real parameters (the phase of a , the phase of b , the relative magnitude between a and b , and the angle θ ). The determinant of such a matrix is:

    det ( U ) = e i θ .

    The sub-group of such elements in U where det ( U ) = 1 is called the special unitary group SU(2).

    The matrix U can also be written in this alternative form:

    U = e i φ [ e i φ 1 cos ⁡ θ e i φ 2 sin ⁡ θ − e − i φ 2 sin ⁡ θ e − i φ 1 cos ⁡ θ ] ,

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

    U = e i φ [ e i ψ 0 0 e − i ψ ] [ cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ ] [ e i Δ 0 0 e − i Δ ] .

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

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

    References

    Unitary matrix Wikipedia
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