Matrix with $A_{ij} = a_i a_j$ for some vector $a\in\mathbb{R}^n$

Question

If $f(x) = g(\langle a,x\rangle)$ for $a,x\in\mathbb{R}^n$ and $g: \mathbb{R}\to\mathbb{R}$, with the usual inner product, then the matrix of second partials is $D^2f(x) = g''(\langle a,x\rangle) A$, where $A$ has components $A_{ij} = a_i a_j$. Is there a name for such a matrix constructed from a vector in this manner, or more generally $A_{ij} = a_i b_j$ for $b\in\mathbb{R}^m$ with $m$ not necessarily equal to $n$?

Answer 1

You can think of it as matrix multiplication or as the Kronecker product. But it's probably more clear to call it the Kronecker product, so that you aren't tempted to the wrong interpretation when $m=n$.

Note that matrix multiplication and the Kronecker product are completely different things except when multiplying an $m\times 1$ by a $1\times n$.