geometric interpretation of element-wise vector multiplication

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.

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 https://en.wikipedia.org/wiki/Ellipsoid, such that the principal semi-axes are just $a_i e_i$. Here $e_i$ is the canonical basis.
• yes of course, the dimension stays the same. Hence the ellipsoid is also in $\mathbb{R}^n$ – humanStampedist Apr 26 '18 at 8:04
• The matrix $A_{ij}$ is square since we only define the diagonal, so the dimension doesn't change. – Tony Apr 26 '18 at 9:09