Outer product

The outer product (denoted as ⊗) of two vectors is a special form of the tensor product or Kronecker product where

$\mathbf {u} \otimes \mathbf {v} =\mathbf {u} \mathbf {v} ^{\mathrm {H} }={\begin{bmatrix}u_{1}\\\vdots \\u_{i}\end{bmatrix}}{\begin{bmatrix}{\overline {v_{1}}}&\cdots &{\overline {v_{j}}}\end{bmatrix}}={\begin{bmatrix}u_{1}{\overline {v_{1}}}&\cdots &u_{1}{\overline {v_{j}}}\\\vdots &\ddots &\vdots \\u_{i}{\overline {v_{1}}}&\cdots &u_{i}{\overline {v_{j}}}\end{bmatrix}}$

v^H represents the conjugate transpose of v. In Einstein summation notation, this is written as

$A_{j}^{i}=u^{i}v_{j}$

The trace of the outer product is the inner product.