nLab
projector

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Idempotents

Contents

Idea

In linear algebra a projector is a linear map e:VVe \colon V \to V that “squares to itself” in that its composition with itself is again itself: ee=ee \circ e = e.

A projector ee leads to a decomposition of the vector space VV that it acts on into a direct sum of its kernel and its image:

Vker(e)im(e). V \simeq ker(e) \oplus im(e) \,.

The notion of projector is the special case of that of idempotent morphism.

In functional analysis, one sometimes requires additionally that this idempotent is in fact self-adjoint; or one can use the slightly different terminology projection operator.

Properties

Projectors relate to the notion of projections in category theory as follows: the existence of the projector P:VVP \colon V \to V canonically induces a decomposition of VV as a direct sum Vker(V)im(V)V \simeq ker(V) \oplus im(V) and in terms of this PP is the composition

P:Vim(v)ker(V)im(V)V P \colon V \simeq im(v) \oplus ker(V)\to im(V) \hookrightarrow V

of the projection (in the sense of maps out of products) out of the direct sum im(V)ker(V)im(V)×ker(V)im(V) \oplus ker(V) \simeq im(V) \times ker(V) followed by the subobject inclusion of im(V)im(V). Hence:

A projector is a projection followed by an inclusion.

Revised on April 6, 2017 00:40:01 by Toby Bartels (108.167.41.14)