(also nonabelian homological algebra)
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In linear algebra a projector is a linear map that “squares to itself” in that its composition with itself is again itself: .
A projector leads to a decomposition of the vector space that it acts on into a direct sum of its kernel and its image:
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.
Projectors relate to the notion of projections in category theory as follows: the existence of the projector canonically induces a decomposition of as a direct sum and in terms of this is the composition
of the projection (in the sense of maps out of products) out of the direct sum followed by the subobject inclusion of . Hence:
A projector is a projection followed by an inclusion.
Last revised on April 6, 2017 at 04:40:01. See the history of this page for a list of all contributions to it.