In linear algebra, a complex square matrix U is invertible and called unitary if its matrix inverse equals its conjugate transpose (U*), satisfying U*U = UU* = I, where I is the identity matrix. In physics and quantum mechanics, the conjugate transpose is called the Hermitian adjoint, denoted by a dagger (†), and unitary matrices satisfy U†U = UU† = I. A unitary matrix is special unitary if its determinant equals 1. For real numbers, the counterpart is an orthogonal matrix. Unitary matrices preserve norms and probability amplitudes, fundamental in quantum theory.
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, ⟨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, as does det ( U ) {\displaystyle \det(U)} .
- 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.1
Equivalent conditions
If U is a square, complex matrix, then the following conditions are equivalent:2
- 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.
Elementary constructions
2 × 2 unitary matrix
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). 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 is3
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.456789
See also
- 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
External links
- Weisstein, Eric W. "Unitary Matrix". MathWorld. Todd Rowland.
- Ivanova, O. A. (2001) [1994], "Unitary matrix", Encyclopedia of Mathematics, EMS Press
- "Show that the eigenvalues of a unitary matrix have modulus 1". Stack Exchange. March 28, 2016.
References
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