where I is the identity matrix.
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, ⟨Ux, Uy⟩ = ⟨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.
- Its eigenspaces are orthogonal.
- U can be written as U = eiH, where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix.
Any square matrix with unit Euclidean norm is the average of two unitary matrices.
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. In other words, U∗U =I.
- The rows of U form an orthonormal basis of with respect to the usual inner product. In other words, U U∗ = I.
- U is an isometry with respect to the usual norm. That is, for all , where .
- U is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of U) with eigenvalues lying on the unit circle.
2 × 2 unitary matrix
The general expression of a 2 × 2 unitary matrix is
which depends on 4 real parameters (the phase of a, the phase of b, the relative magnitude between a and b, and the angle φ). The determinant of such a matrix is
The sub-group of those elements with 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.
Another factorization is
Many other factorizations of a unitary matrix in basic matrices are possible.
- Hermitian matrix
- Matrix decomposition
- Orthogonal group O(n)
- Special orthogonal group SO(n)
- Orthogonal matrix
- Quantum logic gate
- Special Unitary group SU(n)
- Symplectic matrix
- Unitary group U(n)
- Unitary operator
- Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices". Linear and Multilinear Algebra. 50 (4): 321–326. doi:10.1080/03081080290025507.
- Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis. Cambridge University Press. doi:10.1017/9781139020411. ISBN 9781139020411.
- Führ, Hartmut; Rzeszotnik, Ziemowit (2018). "A note on factoring unitary matrices". Linear Algebra and Its Applications. 547: 32–44. doi:10.1016/j.laa.2018.02.017. ISSN 0024-3795.