Outer product: Difference between revisions
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Given two [[vector|vectors]] | |||
$$ | |||
\vec{u}=\sum_i u_i \vec{e}_i,\qquad | |||
\vec{v}=\sum_j v_j \vec{e}_j, | |||
$$ | |||
Their '''outer product''' | |||
$$ | |||
\vec{u} \otimes \vec{v} | |||
= | |||
\sum_{i,j} u_i v_j | |||
$$ | |||
== Example == | |||
For $\vec{u},\vec{v}\in\mathbb{R}^3$ | |||
$$ | |||
\vec{u} \otimes \vec{v} | |||
= | |||
\begin{pmatrix} | |||
u_1 v_1 & u_1 v_2 & u_1 v_3 \\ | |||
u_2 v_1 & u_2 v_2 & u_2 v_3 \\ | |||
u_3 v_1 & u_3 v_2 & u_3 v_3 | |||
\end{pmatrix}. | |||
$$ | |||
Revision as of 13:23, 7 October 2026
Given two vectors
$$ \vec{u}=\sum_i u_i \vec{e}_i,\qquad \vec{v}=\sum_j v_j \vec{e}_j, $$
Their outer product
$$ \vec{u} \otimes \vec{v} = \sum_{i,j} u_i v_j $$
Example
For $\vec{u},\vec{v}\in\mathbb{R}^3$
$$ \vec{u} \otimes \vec{v} = \begin{pmatrix} u_1 v_1 & u_1 v_2 & u_1 v_3 \\ u_2 v_1 & u_2 v_2 & u_2 v_3 \\ u_3 v_1 & u_3 v_2 & u_3 v_3 \end{pmatrix}. $$