Unitary matrix
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
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 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.
- .
- Its eigenspaces are orthogonal.
- U can be written as U = e^{iH}, 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
If U is a square, complex matrix, then the following conditions are equivalent:
- U is unitary.
- U^{∗} is unitary.
- U is invertible with U^{−1} = U^{∗}.
- The columns of U form an orthonormal basis of with respect to the usual inner product.
- The rows of U form an orthonormal basis of with respect to the usual inner product.
- U is an isometry with respect to the usual norm.
- U is a normal matrix with eigenvalues lying on the unit circle.
Elementary constructions
2 × 2 unitary matrix
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 φ_{1} = ψ + Δ and φ_{2} = 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
- Orthogonal matrix
- Hermitian matrix
- Symplectic matrix
- Orthogonal group O(n)
- Special Orthogonal group SO(n)
- Unitary group U(n)
- Special Unitary group SU(n)
- Unitary operator
- Matrix decomposition
- Identity matrix
- Quantum gate
References
- ↑ 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
- Ivanova, O. A. (2001), "Unitary matrix", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4