Editing Bra–ket notation (section)

===Outer products===

A convenient way to define linear operators on a Hilbert space {{math|{{mathcal|H}}}} is given by the [[outer product]]: if {{math|{{bra|''ϕ''}}}} is a bra and {{math|{{ket|''ψ''}}}} is a ket, the outer product
<math display="block"> |\phi\rang \, \lang \psi| </math>
denotes the [[finite-rank operator|rank-one operator]] with the rule 
<math display="block"> \bigl(|\phi\rang \lang \psi|\bigr)(x) = \lang \psi | x \rang |\phi \rang.</math>

For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:
<math display="block"> |\phi \rangle \, \langle \psi | \doteq 
\begin{pmatrix} \phi_1 \\ \phi_2 \\ \vdots \\ \phi_N \end{pmatrix}
\begin{pmatrix} \psi_1^* & \psi_2^* & \cdots & \psi_N^* \end{pmatrix}
= \begin{pmatrix}
\phi_1 \psi_1^* & \phi_1 \psi_2^* & \cdots & \phi_1 \psi_N^* \\
\phi_2 \psi_1^* & \phi_2 \psi_2^* & \cdots & \phi_2 \psi_N^* \\
\vdots & \vdots & \ddots & \vdots \\
\phi_N \psi_1^* & \phi_N \psi_2^* & \cdots & \phi_N \psi_N^* \end{pmatrix}
</math>
The outer product is an {{math|''N'' × ''N''}} matrix, as expected for a linear operator.

One of the uses of the outer product is to construct [[projection operator]]s. Given a ket {{math|{{ket|''ψ''}}}} of norm 1, the orthogonal projection onto the [[Linear subspace|subspace]] spanned by {{math|{{ket|''ψ''}}}} is
<math display="block">|\psi\rangle \, \langle\psi| \,.</math>
This is an [[idempotent]] in the algebra of observables that acts on the Hilbert space.