Outer Vector Product

linear-algebra

Definition

Outer Vector Product

Let $u\in F^m$ and $v\in F^n$ be vectors. The outer vector product of $u$ and $v$ is the matrix

$$u{v}^T$$

whose $(i,j)$-entry is

$$(u{v}^T{)}_{ij}=u_i{v}_j.$$

Equivalently, it is the matrix obtained by multiplying each entry of $u$ with each entry of $v$.

Bra-Ket Notation

The outer product $|x⟩⟨y|$ between a ket-vector $|x⟩$ and a bra-vector $⟨y|$ as a linear operator as follows:

$$\left(|x⟩⟨y|\right)(|z⟩):=⟨y|z⟩|x⟩\forall z\in \mathbb{C}$$