# projector

Given a \(k\) dimensional subspace \(W\) in a \(d\) dimensional vector space \(V\), select an orthonormal basis (using Gram-Schmidt) \(\ket{1}...\ket{d}\) such that \(\ket{1}...\ket{k}\) is an orthonormal basis for \(W\), then
\[
P = \sum_{i=1}^{k}\ket{i}\bra{i}
\]
is a *projector* onto the space \(W\).

Note that \(P\) is *not* a change of basis matrix. The coordinates are still the same. Hitting something with \(P\) just gives the component that lies in \(W\)

## 1. computational basis

We can use \(\ket{0}\bra{0}\) to project onto the component along the \(\ket{0}\) direction. See here for the relevance to multi-qubit gates.

## 2. Properties

*idempotent*: \(P^2 = P\)

## 3. sources

- Nielsen and Chuang pg 70