Tensor product(Outer product) VS Dot Product

• Tensor product(Outer product)

 

The outer product (denoted by ⊗) results in a matrix (or tensor) rather than a scalar.

Let’s say you have two vectors:

 

$\overrightarrow{a}=\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right],\overrightarrow{b}=\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$

$\overrightarrow{a}\otimes \overrightarrow{b}=\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]\otimes \left[\begin{array}{ccc}{b}_1& b_2& b_3\end{array}\right]=\left[\begin{array}{ccc}{a}_1{b}_1& a_1{b}_2& a_1{b}_3\\ a_2{b}_1& a_2{b}_2& a_2{b}_3\\ a_3{b}_1& a_3{b}_2& a_3{b}_3\end{array}\right]$

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• Dot Product

 

It results in a scalar.

 

$\overrightarrow{a}\cdot \overrightarrow{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$