Unitary matrices

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

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

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 (†), so the equation above is written

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

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 ($U^{*}U=UU^{*}$).
• 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.
• $\left|\det(U)\right|=1$. That is, $\det(U)$ will be on the unit circle of the complex plane.
• Its eigenspaces 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.

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

One 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 form is configured so 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).

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

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

By introducing $\ \alpha =\psi +\delta \$ and $\ \beta =\psi -\delta \ ,$ has 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

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

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

See also

• Hermitian matrix and

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

External links

• Weisstein, Eric W. "Unitary Matrix". MathWorld. Todd Rowland.
• Template:SpringerEOM
• "Show that the eigenvalues of a unitary matrix have modulus 1". Stack Exchange. March 28, 2016.