Outer Product - Definition (matrix Multiplication)

Definition (matrix Multiplication)

The outer product u ⊗ v is equivalent to a matrix multiplication uvT, provided that u is represented as a m × 1 column vector and v as a n × 1 column vector (which makes vT a row vector). For instance, if m = 4 and n = 3, then

$\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^T = \begin{bmatrix}u_1 \\ u_2 \\ u_3 \\ u_4\end{bmatrix} \begin{bmatrix}v_1 & v_2 & v_3\end{bmatrix} = \begin{bmatrix}u_1v_1 & u_1v_2 & u_1v_3 \\ u_2v_1 & u_2v_2 & u_2v_3 \\ u_3v_1 & u_3v_2 & u_3v_3 \\ u_4v_1 & u_4v_2 & u_4v_3\end{bmatrix}.$

For complex vectors, it is customary to use the conjugate transpose of v (denoted vH):

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