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Canonical static concept record

Projection operator

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Summary

An idempotent linear operator; an orthogonal projection is also self-adjoint.

Record metadata

Canonical name: Projection operator
Concept ID: projection_operator
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Operator class

Carrier(s)

vector space or Hilbert space

Data

linear operator $P$

Axioms / constraints

$P^2=P$

Canonically induces

decomposition into image and kernel

Notes

An orthogonal projection additionally satisfies $P=P^*$ ; a general projection need not.

Concept sources

Wikipedia — Projection (linear algebra) — Projection (linear algebra) · encyclopedic reference · source ID wp-projection_operator
nLab — projection operator — projection operator · mathematical reference · source ID nlab-projection_operator

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Linear operator Projection operator

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: math_bounded_linear_operator_to_projection_operator
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: idempotence

Authored explanation

An algebraic projection is an everywhere-defined linear operator satisfying $P^2=P$ ; on a Hilbert space, an orthogonal projection additionally satisfies $P=P^*$ and is bounded.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Projection (linear algebra) — Projection (linear algebra) · encyclopedic reference · source ID wp-projection_operator
nLab — projection operator — projection operator · mathematical reference · source ID nlab-projection_operator

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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