Unitary matrix

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.

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]

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.

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]}

• 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.