Editing Positive operator (section)

{{Short description|In mathematics, a linear operator acting on inner product space}}
In [[mathematics]] (specifically [[linear algebra]], [[operator theory]], and [[functional analysis]]) as well as [[physics]], a [[linear operator]] <math>A</math> acting on an [[inner product space]] is called '''positive-semidefinite''' (or ''non-negative'') if, for every <math>x \in \operatorname{Dom}(A)</math>, <math>\langle Ax, x\rangle \in \mathbb{R}</math> and <math>\langle Ax, x\rangle \geq 0</math>, where <math>\operatorname{Dom}(A)</math> is the [[Domain of a function|domain]] of <math>A</math>. Positive-semidefinite operators are denoted as <math>A\ge 0</math>.  The operator is said to be '''positive-definite''', and written <math>A>0</math>, if <math>\langle Ax,x\rangle>0,</math> for all <math>x\in\mathop{\mathrm{Dom}}(A) \setminus \{0\}</math>.<ref>{{harvnb|Roman|2008|loc=p. 250 §10}}</ref>

Many authors define a '''positive operator''' <math>A </math> to be  a [[self-adjoint operator|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 state]]s, via the [[density matrix]] formalism.