Geometrically speaking, vector dot product is interpreted as a projection of one vector on the other, while vector cross product gives another vector that is orthogonal to the multiplied vectors(right?). I am wondering if element-wise vector multiplication has any geometric intuition. I am not a mathematician, so I would appreciate a lay-person friendly explanation.
$\endgroup$1 Answer
$\begingroup$One possible geometric intuition might be the following: Let us call your element-wise multiplication $*$. Then fixate an $a\in\mathbb{R}^n$. In more mathematical terms the map $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ defined by $$x\mapsto a*x$$ is linear and can be represented by a matrix, which would look like $$A_{ij}=\delta_{ij}a_i$$ This leads to the following observation in: If you take the ball $B_1(0)=\{x\in \mathbb{R}^n:\|x\|\leq1\}$ and apply $f$ you obtain an ellipsoid, see also , such that the principal semi-axes are just $a_i e_i$. Here $e_i$ is the canonical basis.
$\endgroup$ 3
