Unitary matrix

In linear algebra, a complex square matrix $U$ is unitary if its conjugate transpose $U *$ is also its inverse, that is, if

U^{*}U=UU^{*}=I,

where $I$ is the identity matrix.

In physics, especially in quantum mechanics, the Hermitian adjoint 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 $U$ 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 ( $U^{*}U=UU^{*}$ ).
$U$ 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=VDV^{*},

where

V

is unitary, and

D

is diagonal and unitary.

$\left|\operatorname {det} (U)\right|=1$ .
Its eigenspaces are orthogonal.
$U$ can be written as $U$ = $e$ ^{$i$ $H$}, 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).

Any 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 is unitary.
U^∗ is unitary.
U is invertible with U⁻¹ = U^∗.
The columns of U form an orthonormal basis of $\mathbb {C} ^{n}$ with respect to the usual inner product. In other words, U^∗U =I.
The rows of U form an orthonormal basis of $\mathbb {C} ^{n}$ with respect to the usual inner product. In other words, U U^∗ = I.
U is an isometry with respect to the usual norm. That is, $\|Ux\|_{2}=\|x\|_{2}$ for all $x\in \mathbb {C} ^{n}$ , where $\|x\|_{2}={\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}$ .
U is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of U) with eigenvalues lying on the unit circle.

Elementary constructions

2 × 2 unitary matrix

The general expression of a 2 × 2 unitary matrix is

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 determinant of such a matrix is

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

The sub-group of those elements $U$ with $\det(U)=1$ is called the special unitary group SU(2).

The matrix $U$ can also be written in this alternative form:

U=e^{i\varphi /2}{\begin{bmatrix}e^{i\varphi _{1}}\cos \theta &e^{i\varphi _{2}}\sin \theta \\-e^{-i\varphi _{2}}\sin \theta &e^{-i\varphi _{1}}\cos \theta \\\end{bmatrix}},

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

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^[3]

U={\begin{bmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \\\end{bmatrix}}{\begin{bmatrix}e^{i\xi }&0\\0&e^{i\zeta }\end{bmatrix}}{\begin{bmatrix}\cos \beta &\sin \beta \\-\sin \beta &\cos \beta \\\end{bmatrix}}.

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

References

^ Li, Chi-Kwong; Poon, Edward (2002). "Additive decomposition of real matrices". Linear and Multilinear Algebra. 50 (4): 321–326. doi:10.1080/03081080290025507.
^ Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis. Cambridge University Press. doi:10.1017/9781139020411. ISBN 9781139020411.
^ Führ, Hartmut; Rzeszotnik, Ziemowit (2018). "A note on factoring unitary matrices". Linear Algebra and Its Applications. 547: 32–44. doi:10.1016/j.laa.2018.02.017. ISSN 0024-3795.

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.

[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] Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis. Cambridge University Press. doi:10.1017/9781139020411. ISBN 9781139020411.

[3] Führ, Hartmut; Rzeszotnik, Ziemowit (2018). "A note on factoring unitary matrices". Linear Algebra and Its Applications. 547: 32–44. doi:10.1016/j.laa.2018.02.017. ISSN 0024-3795.

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