Outer Product - Definition (matrix Multiplication)

Definition (matrix Multiplication)

The outer product uv 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):

Read more about this topic:  Outer Product

Famous quotes containing the word definition:

    According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.
    Ana Castillo (b. 1953)