Unitary matrix

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-by-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:

which depends on 4 real parameters (the phase of ${\displaystyle a}$, the phase of ${\displaystyle b}$, the relative magnitude between ${\displaystyle a}$ and ${\displaystyle b}$, and the angle ${\displaystyle \theta }$). The determinant of such a matrix is:

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