Projection Identities

Formula

An orthogonal projector satisfies

P^2=P, \qquad P^\dagger=P.

Its complement is $Q=I-P$ , with

Q^2=Q, \qquad PQ=QP=0, \qquad P+Q=I.

For a normalized vector $|v\rangle$ ,

P_v=|v\rangle\langle v|.

A projective resolution obeys

P_aP_b=\delta_{ab}P_a, \qquad \sum_aP_a=I,

and a finite-dimensional self-adjoint operator has

A=\sum_a aP_a, \qquad f(A)=\sum_a f(a)P_a.

If commuting projectors $P,Q$ are given, then $PQ$ projects onto their intersection and $P+Q-PQ$ projects onto the sum of their ranges.

Assumptions and Conventions

“Projector” means orthogonal projector on this card.
Infinite sums converge strongly unless a stronger topology is stated.
Continuous spectra require a projection-valued measure, not a naive uncountable sum.

Symbols

Symbol	Meaning
$\operatorname{Ran}P$	selected closed subspace
$\ker P$	orthogonal complement for orthogonal $P$
$P_a$	spectral projector for eigenvalue $a$
$I-P$	complementary projector

Validity and Warnings

Idempotence alone does not imply orthogonal projection; oblique projectors need not be self-adjoint or norm minimizing.
If $[P,Q]\ne0$ , $PQ$ is generally not a projector. The positive sandwich $PQP$ is generally not idempotent either.
In finite dimension, $\operatorname{Tr}P=\operatorname{rank}P$ ; do not transfer this trace formula blindly to infinite rank.
The Born probability $\operatorname{Tr}(\rho P)$ and Lüders update are physical rules in addition to the algebra $P^2=P$ .

Canonical Treatment

Range–kernel geometry, best approximation, spectral construction, noncommuting cases, measurement links, numerical checks, exercises, and references are at Projectors.