where I is the identity matrix.
In physics, especially in quantum mechanics, the Hermitian adjoint of a matrix is denoted by a dagger (†) and the equation above becomes
For any unitary matrix U of finite size, the following hold:
- Given two complex vectors x and y, multiplication by U preserves their inner product; that is, ⟨Ux, Uy⟩ = ⟨x, y⟩.
- U is normal ().
- 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.
- Its eigenspaces are orthogonal.
- U can be written as U = eiH, where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix.
- is unitary.
- is unitary.
- is invertible with .
- The columns of form an orthonormal basis of with respect to the usual inner product. In other words, .
- The rows of form an orthonormal basis of with respect to the usual inner product. In other words, .
- is an isometry with respect to the usual norm. That is, for all , where .
- is a normal matrix (equivalently, there is an orthonormal basis formed by eigenvectors of ) with eigenvalues lying on the unit circle.
2 × 2 unitary matrix
The 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 determinant of such a matrix is
The sub-group of those elements with is called the special unitary group SU(2).
The matrix U can also be written in this alternative form:
which, by introducing φ1 = ψ + Δ and φ2 = ψ − Δ, takes the following factorization:
This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.
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