Good morning, I am looking for a particular product between matrices and vectors defined in the following way:
$ (A\star v)_i=\prod_{j}v_j^{a_{ij}} $
I can give you also and example: consider
$ A=\left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right) \qquad v=\left(\begin{array}{c} a \\ b \end{array}\right) $
The star product gives
$ A\star v=\left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right)\star \left(\begin{array}{c} a \\ b \end{array}\right)=\left(\begin{array}{c} ab^2 \\ a^3b^4 \end{array}\right) $
I met this kind of definition when I was dealing with change of variables for some kind parametrization of the characters of a representation. In particular they used the $\mathfrak{SU}(3)_{\mathbb{C}}$ Cartan matrix as the previous $A$ matrix in such a way they can change the variables for a given representation in variables in the representation of the Cartan matrix. Is this something that I can find in any book? I tried on google but I think that it is not called star product, or anyway it is something so specific that I could not be able to find any proper definition.
Last question: If this kind of change of variables makes sense, is it possible to make it in such a way it is not invertible? I can explain: if I have some variables in a representation, for instance $x_1,x_2,x_3$, it is possible to find a not square matrix in such a way the they can be expressed as one variable $z$? I think that I can reformulate: there are not square Cartan matrices?
If $M$ is a monoid and $k$ is a commutative ring, then we can consider the monoid algebra $k[M]$. When $M = \mathbb{N}^n$ this recovers the polynomial ring over $k$ in $n$ variables, and when $M = \mathbb{Z}^n$ this recovers the Laurent polynomial ring over $k$ in $n$ variables.
If $f : M \to N$ is a homomorphism of monoids, we get an induced homomorphism of monoid algebras $k[f] : k[M] \to k[N]$.
Finally, if $M = \mathbb{N}^m, N = \mathbb{N}^n$, then homomorphisms $M \to N$ correspond exactly to $n \times m$ matrices over $\mathbb{N}$, which compose as matrices do, and similarly for $\mathbb{Z}^m$ and $\mathbb{Z}^n$. These induce homomorphisms on polynomial resp. Laurent polynomial algebras which correspond to your operation.