Outer Product

linear-algebra

Definition

Outer Product

Let . The outer product of column vectors and is the matrix

where denotes the conjugate transpose of . Over , conjugation is trivial, so .

Einstein Notation:

Bra-Ket Notation

Let be a finite-dimensional complex Hilbert space. In bra-ket notation, the outer product is the linear operator

This follows from associativity of matrix multiplication:

Here is a scalar. Thus the operator extracts the component of along and returns that scalar multiple of .