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 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.
- 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.
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 introduced in :
Many other factorizations of a unitary matrix in basic matrices are possible.
- 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
- Quantum gate
- Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices". Linear and Multilinear Algebra. 50 (4): 321–326. doi:10.1080/03081080290025507.
- 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.
- Weisstein, Eric W. "Unitary Matrix". MathWorld.
- Ivanova, O. A. (2001) , "Unitary matrix", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- “Show that the eigenvalues of a unitary matrix have modulus 1” on Mathematics Stack Exchange