Unitary matrix

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

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 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

$U=VDV^{*}\;$

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.
• U can be written as U=e^iH where e indicates matrix exponential, i is the imaginary unit and H is an Hermitian matrix.

Equivalent conditions

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

2x2 Unitary matrix

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

The general expression of 3x3 unitary matrix is:

$U={\begin{bmatrix}1&0&0\\0&e^{j\varphi _{4}}&0\\0&0&e^{j\varphi _{5}}\end{bmatrix}}K{\begin{bmatrix}e^{j\varphi _{1}}&0&0\\0&e^{j\varphi _{2}}&0\\0&0&e^{j\varphi _{3}}\end{bmatrix}}$

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

See also

• Orthogonal matrix
• Hermitian matrix
• Symplectic matrix
• Unitary group
• Special unitary group
• Unitary operator
• Matrix decomposition
• Identity matrix
• Quantum gate

External links

• Rowland, Todd. "Unitary Matrix". MathWorld. {{cite web}}: More than one of |author= and |last= specified (help)
• Ivanova, O. A. (2001) [1994], "Unitary matrix", Encyclopedia of Mathematics, EMS Press