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 (†), so the equation above is written
A complex matrix U is special unitary if it is unitary and its matrix determinant equals 1.
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 where V is unitary, and D is diagonal and unitary.
. That is, will be on the unit circle of the complex plane.
One general expression of a 2 × 2 unitary matrix is
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
The sub-group of those elements with is called the special unitary group SU(2).
Among several alternative forms, the matrix U can be written in this form:
where and above, and the angles can take any values.
By introducing and has the following factorization:
This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2orthogonal matrices of angle θ.
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