Outer Product

linear-algebra

Definition

Outer Product

Let $F\in \{\mathbb{R},\mathbb{C}\}$. The outer product of column vectors $u\in F^m$ and $v\in F^n$ is the $m\times n$ matrix

$$u{v}^{†},(u{v}^{†}{)}_{ij}=u_i\overline{v_j},$$

where $v^{†}$ denotes the conjugate transpose of $v$. Over $\mathbb{R}$, conjugation is trivial, so $(u{v}^T{)}_{ij}=u_i{v}_j$.

Einstein Notation: $(u{v}^{†}{)}_{ij}=u_i{v}_j$

Bra-Ket Notation

Let $\mathcal{H}$ be a finite-dimensional complex Hilbert space. In bra-ket notation, the outer product $|x⟩⟨y|$ is the linear operator

$$|x⟩⟨y|:|z⟩\mapsto ⟨y|z⟩|x⟩.$$

This follows from associativity of matrix multiplication:

$$\begin{array}{rl}(|x⟩⟨y|)|z⟩& =|x⟩(⟨y||z⟩)\\ & =|x⟩⟨y|z⟩\\ & =⟨y|z⟩|x⟩.\end{array}$$

Here $⟨y||z⟩=⟨y|z⟩\in \mathbb{C}$ is a scalar. Thus the operator extracts the component of $|z⟩$ along $|y⟩$ and returns that scalar multiple of $|x⟩$.