Unitary matrix

For matrices with orthogonality over the real number field, see orthogonal matrix. For the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1, see unitarity.

In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U⁻¹ equals its conjugate transpose U^*, that is, if

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

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 (⁠${\displaystyle \dagger }$⁠), so the equation above is written

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

A complex matrix U is special unitary if it is unitary and its matrix determinant equals 1.

For real numbers, the 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.
• The eigenvalues of ${\displaystyle U}$ lie on the unit circle, as does ${\displaystyle \det(U)}$.
• The eigenspaces of ${\displaystyle U}$ are orthogonal.
• U can be written as U = e^iH, 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).

Every 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

One 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 form is configured so 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).

Among several alternative forms, the matrix U can be written in this form: $${\displaystyle \ U=e^{i\varphi /2}{\begin{bmatrix}e^{i\alpha }\cos \theta &e^{i\beta }\sin \theta \\-e^{-i\beta }\sin \theta &e^{-i\alpha }\cos \theta \\\end{bmatrix}}\ ,}$$

where ${\displaystyle \ e^{i\alpha }\cos \theta =a\ }$ and ${\displaystyle \ e^{i\beta }\sin \theta =b\ ,}$ above, and the angles ${\displaystyle \ \varphi ,\alpha ,\beta ,\theta \ }$ can take any values.

By introducing ${\displaystyle \ \alpha =\psi +\delta \ }$ and ${\displaystyle \ \beta =\psi -\delta \ ,}$ has 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]

$${\displaystyle U={\begin{bmatrix}\cos \rho &-\sin \rho \\\sin \rho &\;\cos \rho \\\end{bmatrix}}{\begin{bmatrix}e^{i\xi }&0\\0&e^{i\zeta }\end{bmatrix}}{\begin{bmatrix}\;\cos \sigma &\sin \sigma \\-\sin \sigma &\cos \sigma \\\end{bmatrix}}~.}$$

Many other factorizations of a unitary matrix in basic matrices are possible.^{[4][5][6][7][8][9]}

See also

• Hermitian matrix
• Skew-Hermitian matrix
• Matrix decomposition
• Orthogonal group O(n)
• Special orthogonal group SO(n)
• Orthogonal matrix
• Semi-orthogonal matrix
• Quantum logic gate
• Special Unitary group SU(n)
• Symplectic matrix
• Unitary group U(n)
• Unitary operator