Unitary matrix

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In mathematics, a complex square matrix U is unitary if

U^* U = UU^* = I \,

where I is the identity matrix and U* is the conjugate transpose of U. In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger (†) and the equation above becomes

U^\dagger U = UU^\dagger = I. \,

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

Properties[edit]

For any unitary matrix U, the following hold:

  • Given two complex vectors x and y, multiplication by U preserves their inner product; that is,
\langle Ux, Uy \rangle = \langle x, y \rangle.
U = VDV^*\;
where V is unitary and D is diagonal and unitary.
  • |\det(U)|=1.
  • Its eigenspaces are orthogonal.
  • For any nonnegative integer n, the set of all n by n unitary matrices with matrix multiplication forms a group, called the unitary group U(n).
  • Any square matrix with unit Euclidean norm is the average of two unitary matrices.[1]
  • U can be written as U=eiH where e indicates matrix exponential, i is the imaginary unit and H is an Hermitian matrix.

Equivalent conditions[edit]

If U is a square, complex matrix, then the following conditions are equivalent:

  1. U is unitary.
  2. U* is unitary.
  3. U is invertible with U –1=U*.
  4. The columns of U form an orthonormal basis of \mathbb{C}^n with respect to the usual inner product.
  5. The rows of U form an orthonormal basis of \mathbb{C}^n with respect to the usual inner product.
  6. U is an isometry with respect to the usual norm.
  7. U is a normal matrix with eigenvalues lying on the unit circle.

Elementary constructions[edit]

2x2 Unitary matrix[edit]

The general expression of a 2x2 unitary matrix is:

U = 
e^{i\varphi}\begin{bmatrix}
a & b \\
-b^* & a^* \\
\end{bmatrix},\qquad |a|^2 + |b|^2 = 1 ,

which depends on 4 real parameters. The determinant of such a matrix is:


\det(U)=e^{i2\varphi} .

If φ=0, the group created by U is called special unitary group SU(2).

Matrix U can also be written in this alternative form:

U = 
e^{i\varphi}\begin{bmatrix}
\cos \theta e^{i\varphi_1} & \sin \theta e^{i\varphi_2}\\
-\sin \theta e^{-i\varphi_2}& \cos \theta e^{-i\varphi_1}\\
\end{bmatrix} ,

which, by introducing φ1 = ψ + Δ and φ2 = ψ - Δ, takes the following factorization:

U = 
e^{i\varphi}\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 2x2 unitary matrices and 2x2 orthogonal matrices of angle θ.

Many other factorizations of a unitary matrix in basic matrices are possible.

3x3 Unitary matrix[edit]

The general expression of 3x3 unitary matrix is:[2]

U = \begin{bmatrix}
1 & 0 & 0 \\
0 & e^{j\varphi_4} & 0 \\
0 & 0 & e^{j\varphi_5}
\end{bmatrix} 
K
\begin{bmatrix}
e^{j\varphi_1} & 0 & 0 \\
0 & e^{j\varphi_2} & 0 \\
0 & 0 & e^{j\varphi_3}
\end{bmatrix}

where φn, n=1,...,5 are arbitrary real numbers, while K is the Cabibbo–Kobayashi–Maskawa matrix.

See also[edit]

References[edit]

  1. ^ Li, Chi-Kwong; Poon, Edward (2002). "Additive Decomposition of Real Matrices". Linear and Multilinear Algebra 50 (4): 321–326. doi:10.1080/03081080290025507. 
  2. ^ A. Rašin (1997). "Diagonalization of Quark Mass Matrices and the Cabibbo-Kobayashi-Maskawa Matrix". arXiv:9708216 [hep-ph].

External links[edit]