Unitary matrix

Equivalent conditions

If U is a square, complex matrix, then the following conditions are equivalent:

1. ${\displaystyle U}$ is unitary.
2. ${\displaystyle U^{\ast }}$ is unitary.
3. ${\displaystyle U}$ is invertible with ${\displaystyle U^{-1}=U^{\ast }}$.
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, ${\displaystyle U^{\ast }U=I}$.
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, ${\displaystyle U{U}^{\ast }=I}$.
6. ${\displaystyle U}$ is an isometry with respect to the usual norm. That is, ${\displaystyle \Vert Ux{\Vert }_2=\Vert x{\Vert }_2}$ for all ${\displaystyle x\in {\mathbb{C}}^n}$, where $\Vert x{\Vert }_2=\sqrt{\sum _{i=1}^n|x_i{|}^2}$.
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 $${\displaystyle U=\left[\begin{array}{cc}a& b\\ -e^{i\phi }{b}^{\ast }& e^{i\phi }{a}^{\ast }\end{array}\right],{\left|a\right|}^2+{\left|b\right|}^2=1,}$$

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) and * is the complex conjugate. The form is configured so the determinant of such a matrix is $${\displaystyle det(U)=e^{i\phi }.}$$

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: $${\displaystyle U=e^{i\phi /2}\left[\begin{array}{cc}{e}^{i\alpha }\cos \theta & e^{i\beta }\sin \theta \\ -e^{-i\beta }\sin \theta & e^{-i\alpha }\cos \theta \end{array}\right],}$$

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

By introducing ${\displaystyle \alpha =\psi +\delta }$ and ${\displaystyle \beta =\psi -\delta ,}$ has the following factorization:

$${\displaystyle U=e^{i\phi /2}\left[\begin{array}{cc}{e}^{i\psi }& 0\\ 0& e^{-i\psi }\end{array}\right]\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{cc}{e}^{i\delta }& 0\\ 0& e^{-i\delta }\end{array}\right].}$$

This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.

Another factorization is

$${\displaystyle U=\left[\begin{array}{cc}\cos \rho & -\sin \rho \\ \sin \rho & \cos \rho \end{array}\right]\left[\begin{array}{cc}{e}^{i\xi }& 0\\ 0& e^{i\zeta }\end{array}\right]\left[\begin{array}{cc}\cos \sigma & \sin \sigma \\ -\sin \sigma & \cos \sigma \end{array}\right].}$$

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