Unitary Matrices
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Unitary Matrices

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

${\displaystyle U^{*}U=UU^{*}=I,}$

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

${\displaystyle U^{\dagger }U=UU^{\dagger }=I.}$

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 (${\displaystyle U^{*}U=UU^{*}}$).
• 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
${\displaystyle U=VDV^{*},}$
where V is unitary, and D is diagonal and unitary.
• ${\displaystyle \left|\operatorname {det} (U)\right|=1}$.
• Its eigenspaces 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).

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]

1. U is unitary.
2. U* is unitary.
3. U is invertible with .
4. The columns of U form an orthonormal basis of ${\displaystyle \mathbb {C} ^{n}}$ with respect to the usual inner product. In other words, U*U =I.
5. The rows of U form an orthonormal basis of ${\displaystyle \mathbb {C} ^{n}}$ with respect to the usual inner product. In other words, U U* = I.
6. U is an isometry with respect to the usual norm. That is, ${\displaystyle \|Ux\|_{2}=\|x\|_{2}}$ for all ${\displaystyle x\in \mathbb {C} ^{n}}$, where ${\displaystyle \|x\|_{2}={\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}}$.
7. 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 unitary matrix is

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

which depends on 4 real parameters (the phase of a, the phase of b, the relative magnitude between a and b, and the angle ?). The determinant of such a matrix is

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

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

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

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

which, by introducing and , takes the following factorization:

${\displaystyle U=e^{i\varphi /2}{\begin{bmatrix}e^{i\psi }&0\\0&e^{-i\psi }\end{bmatrix}}{\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}e^{i\Delta }&0\\0&e^{-i\Delta }\end{bmatrix}}.}$

This expression highlights the relation between unitary matrices and orthogonal matrices of angle ?.

Another factorization is[3]

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

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

## References

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.