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

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[edit]

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,
.
where V is unitary and D is diagonal and unitary.
  • .
  • 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.[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 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[edit]

2 × 2 unitary matrix[edit]

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

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

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

The 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 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.

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

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]