In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator
acting on an inner product space is called positive-semidefinite (or non-negative) if, for every
Positive-semidefinite operators are denoted as
The operator is said to be positive-definite, and written
[1] Many authors define a positive operator
to be a self-adjoint (or at least symmetric) non-negative operator.
We show below that for a complex Hilbert space the self adjointness follows automatically from non-negativity.
For a real Hilbert space non-negativity does not imply self adjointness.
In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism.
to be anti-linear on the first argument and linear on the second and suppose that
is positive and symmetric, the latter meaning that
Then the non negativity of for all complex
{\displaystyle \mathop {\text{Im}} A\perp \mathop {\text{Ker}} A.}
the polarization identity and the fact that
for positive operators, show that
In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space
to be an operator of rotation by an acute angle
φ ∈ ( − π
x ‖ ‖ x ‖ cos φ > 0 ,
In our case, the equality of domains holds because
A natural partial ordering of self-adjoint operators arises from the definition of positive operators.
if the following hold: It can be seen that a similar result as the Monotone convergence theorem holds for monotone increasing, bounded, self-adjoint operators on Hilbert spaces.
[2] The definition of a quantum system includes a complex separable Hilbert space
of positive trace-class operators
is the set of states.
is called a state or a density operator.
is called a pure state.
(Since each pure state is identifiable with a unit vector
some sources define pure states to be unit elements from
States that are not pure are called mixed.