Projection operator

A projection operator $𝑃 : 𝑉 \to 𝑉$ is an Idempotent linear endomorphism, i.e. an operator whose restriction $𝑃 ↾ 𝑃 𝑉$ to its image is the identity. #m/def/linalg

𝑃 2 𝑣 = 𝑃 𝑣

The geometric intuition is that a projection operator projects vectors into the subspace $𝑃 𝑉$ .

Properties

In bra-ket notation the projection operator onto the space spanned by normalized $| 𝑣 ⟩$ is $| 𝑣 ⟩ ⟨ 𝑣 |$ .

#state/tidy | #lang/en | #SemBr