Projection Operators

Vectors in $R^{3}$

$\vec{a} \cdot \vec{b} = a^{x} b^{x} + a^{y} b^{y} + a^{z} b^{z} = a^{i} b^{i} = a b \cos θ = a_{| |} b = a b_{| |}$ .

They pick out the components of a vector to a subspace of $V$ .

$P^{†} P = P$ .

$P^{2} = P$
$P^{†} = P$
$P_{1} + P_{2}$ is a projection operator iff $P_{1} P_{2} = P_{2} P_{1} = 0$ (Orthogonal projections)

$W^{⊥} = {| w^{⊥} ⟩ : ⟨ w^{⊥} | w ⟩ \forall | w ⟩ \in W}$ .

$| v ⟩ = | v_{w} ⟩ + | v_{w}^{⊥} ⟩$
$V = W \oplus W^{⊥}$ , $\oplus$ is called the direct sum.
If $P_{w}$ projects onto $W$
- $P_{w} | v ⟩ = | w ⟩ \in W$
- $P_{w} | w ⟩ = | w ⟩ \forall | w ⟩ \in W$
then $P_{w}^{⊥} = I - P_{w}$ projects onto $W$ .
Then if ${P_{i}}$ is a set of orthogonal projections, $P_{i} P_{j} = P_{i} δ_{j}^{i}$ , then $P = \sum_{i} P_{i}$ projects onto the `span of $P_{i}$ ’.

If ${P_{i}}$ is an orthonormal basis for $V$ then

$P_{i} = | i ⟩ ⟨ i |$ projects onto the $i$-th basis direction
$P_{i} + P_{j} = | i ⟩ ⟨ i | + | j ⟩ ⟨ j |$ projects onto the subspace spanned by the $i$ and $j$-th basis vectors
$\sum_{i} P_{i} = I$

Proof.

$P_{i} | a ⟩ = | i ⟩ ⟨ i | a ⟩ = a^{i} | i ⟩$ .
$(P_{i} + P_{j}) | a ⟩ = | i ⟩ ⟨ i | a ⟩ + | j ⟩ ⟨ j | a ⟩ = a^{i} | i ⟩ + a^{j} | j ⟩$ .
$[\sum_{i} P_{i}] | a ⟩ = [\sum_{i} P_{i} | a ⟩] = \sum_{i} a^{i} | i ⟩ = | a ⟩$ .

$P_{i} = | i ⟩ ⟨ i |$ . $\sum_{n} P_{n} = I$ iff ${| i ⟩}$ is orthonormal basis (spanning, linearly independent, orthogonal, normal).

$I | a ⟩ = \sum_{n} P_{n} | a ⟩ = a^{n} | n ⟩$ .
$A = I A I = \sum_{n} P_{n} A \sum_{n} P_{m} = \sum_{n} \sum_{m} | n ⟩ ⟨ m | ⟨ n | A | m ⟩$ .
$A_{j}^{i} = ⟨ i | A | j ⟩ = \sum_{n} \sum_{m} ⟨ i | n ⟩ ⟨ m | j ⟩ ⟨ n | A | m ⟩ = \sum_{n} \sum_{m} δ_{n}^{i} δ_{j}^{m} ⟨ n | A | m ⟩ = ⟨ i | A | j ⟩$ .

$A = \sum_{n} A^{n} P_{n}$