In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if
U ∗ U = U U ∗ = I , {\displaystyle U^{*}U=UU^{*}=I,}
where I is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (⁠ † {\displaystyle \dagger } ⁠), so the equation above is written
U † U = U U † = I . {\displaystyle U^{\dagger }U=UU^{\dagger }=I.}
A complex matrix U is special unitary if it is unitary and its matrix determinant equals 1.
For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve the normalization of state vectors and the inner products between them.
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 ∗ U = U U ∗ {\displaystyle U^{*}U=UU^{*}} ).
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 U = V D V ∗ , {\displaystyle U=VDV^{*},} where V is unitary, and D is diagonal and unitary.
The eigenvalues of U {\displaystyle U} lie on the unit circle. That is, if the complex number λ is an eigenvalue of U then |λ| = 1.
| det ( U ) | = 1 {\displaystyle |\det(U)|=1}
The eigenspaces of U {\displaystyle U} 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.
For any nonnegative integer n, the set of all n × n unitary matrices with matrix multiplication forms a group, called the unitary group U(n).
Every 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 {\displaystyle U} is unitary.
U ∗ {\displaystyle U^{*}} is unitary.
U {\displaystyle U} is invertible with U − 1 = U ∗ {\displaystyle U^{-1}=U^{*}} .
The columns of U {\displaystyle U} form an orthonormal basis of C n {\displaystyle \mathbb {C} ^{n}} with respect to the usual inner product. In other words, U ∗ U = I {\displaystyle U^{*}U=I} .
The rows of U {\displaystyle U} form an orthonormal basis of C n {\displaystyle \mathbb {C} ^{n}} with respect to the usual inner product. In other words, U U ∗ = I {\displaystyle UU^{*}=I} .
U {\displaystyle U} is an isometry with respect to the usual norm. That is, ‖ U x ‖ 2 = ‖ x ‖ 2 {\displaystyle \|Ux\|_{2}=\|x\|_{2}} for all x ∈ C n {\displaystyle x\in \mathbb {C} ^{n}} , where ‖ x ‖ 2 = ∑ i = 1 n | x i | 2 {\textstyle \|x\|_{2}={\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}} .
U {\displaystyle U} is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of U {\displaystyle U} ) with eigenvalues lying on the unit circle.
One general expression of a 2 × 2 unitary matrix is U = [ a b − e i φ b ∗ e i φ a ∗ ] , | a | 2 + | b | 2 = 1 , {\displaystyle U={\begin{bmatrix}a&b\\-e^{i\varphi }b^{*}&e^{i\varphi }a^{*}\\\end{bmatrix}},\qquad \left|a\right|^{2}+\left|b\right|^{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 φ) and * is the complex conjugate. The form is configured so the determinant of such a matrix is det ( U ) = e i φ . {\displaystyle \det(U)=e^{i\varphi }~.}
The sub-group of those elements U {\displaystyle U} with det ( U ) = 1 {\displaystyle \det(U)=1} is called the special unitary group SU(2).
Among several alternative forms, the matrix U can be written in this form: U = e i φ / 2 [ e i α cos ⁡ θ e i β sin ⁡ θ − e − i β sin ⁡ θ e − i α cos ⁡ θ ] , {\displaystyle \ U=e^{i\varphi /2}{\begin{bmatrix}e^{i\alpha }\cos \theta &e^{i\beta }\sin \theta \\-e^{-i\beta }\sin \theta &e^{-i\alpha }\cos \theta \\\end{bmatrix}}\ ,}
where e i α cos ⁡ θ = a {\displaystyle e^{i\alpha }\cos \theta =a} and e i β sin ⁡ θ = b , {\displaystyle e^{i\beta }\sin \theta =b,} above, and the angles φ , α , β , θ {\displaystyle \varphi ,\alpha ,\beta ,\theta } can take any values.
By introducing α = ψ + δ {\displaystyle \alpha =\psi +\delta } and β = ψ − δ , {\displaystyle \beta =\psi -\delta ,} has the following factorization:
U = e i φ / 2 [ e i ψ 0 0 e − i ψ ] [ cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ ] [ e i δ 0 0 e − i δ ] . {\displaystyle U=e^{i\varphi /2}{\begin{bmatrix}e^{i\psi }&0\\0&e^{-i\psi }\end{bmatrix}}{\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}e^{i\delta }&0\\0&e^{-i\delta }\end{bmatrix}}~.}
This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.
Another factorization is
U = [ cos ⁡ ρ − sin ⁡ ρ sin ⁡ ρ cos ⁡ ρ ] [ e i ξ 0 0 e i ζ ] [ cos ⁡ σ sin ⁡ σ − sin ⁡ σ cos ⁡ σ ] . {\displaystyle U={\begin{bmatrix}\cos \rho &-\sin \rho \\\sin \rho &\;\cos \rho \\\end{bmatrix}}{\begin{bmatrix}e^{i\xi }&0\\0&e^{i\zeta }\end{bmatrix}}{\begin{bmatrix}\;\cos \sigma &\sin \sigma \\-\sin \sigma &\cos \sigma \\\end{bmatrix}}~.}
Many other factorizations of a unitary matrix in basic matrices are possible.
Hermitian matrix
Skew-Hermitian matrix
Matrix decomposition
Orthogonal group O(n)
Special orthogonal group SO(n)
Orthogonal matrix
Semi-orthogonal matrix
Quantum logic gate
Special Unitary group SU(n)
Symplectic matrix
Unitary group U(n)
Unitary operator
Unitary transformation
