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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}

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

A^i_j = u^i v_j

The trace of the outer product is the inner product.

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