nLab projector

Idea

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

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

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

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 $P \colon V \to V$ canonically induces a decomposition of $V$ as a direct sum $V \simeq ker(V) \oplus im(V)$ and in terms of this $P$ is the composition

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

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

A projector is a projection followed by an inclusion.

Related concepts

• projection