Alphabeta MathHow statement and proof provenance work
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Non-negative and positive operators
Definition
Let be a finite-dimensional real or complex inner product space and let be linear. The operator is non-negative when is self-adjoint in the sense of Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space and
for every .
The operator is positive when is self-adjoint and
for every nonzero .
Depends on
Used by
- FALSE: If <Tv,v> is nonnegative for every v, then T is automatically self-adjoint False statement
- Over the reals, non-negative and positive operators correspond exactly to positive semidefinite and positive definite symmetric forms Proposition
- A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S Theorem
Dependency tree · two levels
2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Sheldon Axler, Linear Algebra Done Right, fourth edition (standard reference, not scraped)