Unitary matrix

In mathematics, a complex square matrix U is unitary if its conjugate transpose U∗ is also its inverse—that is, if In mathematics, a complex square matrix U is unitary if its conjugate transpose U∗ is also its inverse—that is, if where I is the identity matrix. In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger (†) and the equation above becomes The real 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: 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). Any square matrix with unit Euclidean norm is the average of two unitary matrices. If U is a square, complex matrix, then the following conditions are equivalent: The general expression of a 2 × 2 unitary matrix is: