Lesson plan
Positive operator
In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or non-negative) if, for every x \in \operatorname{Dom}(A) , \langle Ax, x\rangle \in \mathbb{R} and \langle Ax, x\rangle \geq 0 , where \operatorname{Dom}(A) is the domain of A . Positive-semidefinite operators are denoted as A\ge 0 . The operator is said to be positive-definite, and written A>0 , if \langle Ax,x\rangle>0 for all x\in\mathop{\mathrm{Dom}}(A) \setminus \{0\} . Many authors define a positive operator A 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.
Article summary
In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or non-negative) if, for every x \in \operatorname{Dom}(A) , \langle Ax, x\rangle \in \mathbb{R} and \langle Ax, x\rangle \geq 0 , where \operatorname{Dom}(A) is the domain of A . Positive-semidefinite operators are denoted as A\ge 0 . The operator is said to be positive-definite, and written A>0 , if \langle Ax,x\rangle>0 for all x\in\mathop{\mathrm{Dom}}(A) \setminus \{0\} . Many authors define a positive operator A 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.
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Discussion questions
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