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Unitary matrix

## Summary

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

${\displaystyle U^{*}U=UU^{*}=UU^{-1}=I,}$

where I is the identity matrix.

In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written

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

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.
• ${\displaystyle \left|\det(U)\right|=1}$ .
• 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. ${\displaystyle U}$  is unitary.
2. ${\displaystyle U^{*}}$  is unitary.
3. ${\displaystyle U}$  is invertible with ${\displaystyle U^{-1}=U^{*}}$ .
4. The columns of ${\displaystyle U}$  form an orthonormal basis of ${\displaystyle \mathbb {C} ^{n}}$  with respect to the usual inner product. In other words, ${\displaystyle U^{*}U=I}$ .
5. The rows of ${\displaystyle U}$  form an orthonormal basis of ${\displaystyle \mathbb {C} ^{n}}$  with respect to the usual inner product. In other words, ${\displaystyle UU^{*}=I}$ .
6. ${\displaystyle U}$  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 ${\textstyle \|x\|_{2}={\sqrt {\sum _{i=1}^{n}|x_{i}|^{2}}}}$ .
7. ${\displaystyle U}$  is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of ${\displaystyle U}$ ) with eigenvalues lying on the unit circle.

## Elementary constructions

### 2 × 2 unitary matrix

The general expression of a 2 × 2 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 φ). 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 φ1 = ψ + Δ and φ2 = ψ − Δ, 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 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.

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.[4][5][6][7]