Unitary matrix

For any unitary matrix of finite size, the following hold: • Given two complex vectors and , multiplication by preserves their inner product; that is, . • is normal (U^* U = UU^*). • is diagonalizable; that is, is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, has a decomposition of the form U = VDV^*, where is unitary, and is diagonal and unitary. • The eigenvalues of U lie on the unit circle, as does \det(U). • The eigenspaces of U are orthogonal. • can be written as , where indicates the matrix exponential, is the imaginary unit, and is a Hermitian matrix. For any nonnegative integer , the set of all unitary matrices with matrix multiplication forms a group, called the unitary group . 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: • U is unitary. • U^* is unitary. • U is invertible with U^{-1} = U^*. • The columns of U form an orthonormal basis of \Complex^n with respect to the usual inner product. In other words, U^*U = I. • The rows of U form an orthonormal basis of \Complex^n with respect to the usual inner product. In other words, UU^* = I. • U is an isometry with respect to the usual norm. That is, \|Ux\|_2 = \|x\|_2 for all x \in \Complex^n, where \|x\|_2 = \sqrt{\sum_{i=1}^n |x_i|^2}. • 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 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 , the phase of , the relative magnitude between and , and the angle ) and * is the complex conjugate. 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 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 unitary matrices and orthogonal matrices of angle . 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==