Complex matrix whose conjugate transpose equals its inverse
For matrices with orthogonality over the real number field, see
orthogonal matrix . For the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1, see
unitarity .
Inlinear algebra , aninvertible complex square matrix U isunitary if itsmatrix inverse U −1 conjugate transpose U *
U ∗ U = U U ∗ = I , {\displaystyle U^{*}U=UU^{*}=I,}
whereI is theidentity matrix .
Inphysics , especially inquantum mechanics , the conjugate transpose is referred to as theHermitian adjoint of a matrix and is denoted by adagger († {\displaystyle \dagger } ), so the equation above is written
U † U = U U † = I . {\displaystyle U^{\dagger }U=UU^{\dagger }=I.}
A complex matrixU isspecial unitary if it is unitary and itsmatrix determinant equals1 .
Forreal numbers , the analogue of a unitary matrix is anorthogonal matrix . Unitary matrices have significant importance in quantum mechanics because they preservenorms , and thus,probability amplitudes .
For any unitary matrixU of finite size, the following hold:
Given two complex vectorsx y U preserves theirinner product ; that is,⟨U x ,U y ⟩ = ⟨x ,y ⟩ . U isnormal (U ∗ U = U U ∗ {\displaystyle U^{*}U=UU^{*}} U isdiagonalizable ; that is,U isunitarily similar to a diagonal matrix, as a consequence of thespectral theorem . Thus,U has a decomposition of the formU = V D V ∗ , {\displaystyle U=VDV^{*},} V is unitary, andD is diagonal and unitary.Theeigenvalues ofU {\displaystyle U} unit circle , as doesdet ( U ) {\displaystyle \det(U)} Theeigenspaces ofU {\displaystyle U} U can be written asU =e iH e indicates thematrix exponential ,i is the imaginary unit, andH is aHermitian matrix .For any nonnegativeinteger n n × n group , called theunitary group U(n ) .
Every square matrix with unit Euclidean norm is the average of two unitary matrices.[ 1]
Equivalent conditions [ edit ] IfU is a square, complex matrix, then the following conditions are equivalent:[ 2]
U {\displaystyle U} U ∗ {\displaystyle U^{*}} U {\displaystyle U} U − 1 = U ∗ {\displaystyle U^{-1}=U^{*}} The columns ofU {\displaystyle U} orthonormal basis ofC n {\displaystyle \mathbb {C} ^{n}} U ∗ U = I {\displaystyle U^{*}U=I} The rows ofU {\displaystyle U} C n {\displaystyle \mathbb {C} ^{n}} U U ∗ = I {\displaystyle UU^{*}=I} U {\displaystyle U} isometry with respect to the usual norm. That is,‖ U x ‖ 2 = ‖ x ‖ 2 {\displaystyle \|Ux\|_{2}=\|x\|_{2}} x ∈ C n {\displaystyle x\in \mathbb {C} ^{n}} ‖ x ‖ 2 = ∑ i = 1 n | x i | 2 {\textstyle \|x\|_{2}={\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}} U {\displaystyle U} normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors ofU {\displaystyle U} Elementary constructions [ edit ] [ edit ] One general expression of a2 × 2 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 ofa , the phase ofb , the relative magnitude betweena andb , and the angleφ ) and * is thecomplex conjugate . The form is configured so thedeterminant of such a matrix isdet ( U ) = e i φ . {\displaystyle \det(U)=e^{i\varphi }~.}
The sub-group of those elementsU {\displaystyle U} det ( U ) = 1 {\displaystyle \det(U)=1} special unitary group SU(2).
Among several alternative forms, the matrixU 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}}\ ,}
wheree i α cos θ = a {\displaystyle e^{i\alpha }\cos \theta =a} e i β sin θ = b , {\displaystyle e^{i\beta }\sin \theta =b,} φ , α , β , θ {\displaystyle \varphi ,\alpha ,\beta ,\theta }
By introducingα = ψ + δ {\displaystyle \alpha =\psi +\delta } β = ψ − δ , {\displaystyle \beta =\psi -\delta ,}
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 between2 × 2 2 × 2 orthogonal matrices of angleθ .
Another factorization is[ 3]
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.[ 4] [ 5] [ 6] [ 7] [ 8] [ 9]
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