Operators

Hilbert Space

The Hilbert Space is an inner product space, which is complete with respect to the norm defined by the inner product.

$U^\dagger$ is the adjoint of $U$, and in matrix representation is given by $(U^*)^T$, where $^*$ marks the complex conjugate and $^T$ the transpose.
An operator (matrix) $U$ is said to be normal if $U^\dagger U = U U^\dagger$.
- A normal operator is diagonizable (spectral decomposition)
An operator (matrix) $U$ is said to be unitary if $U^\dagger U = I$.
An operator (matrix) $U$ is said to be hermitian if $U = U^\dagger$.
- A hermitian operator is normal.