Is there some nice way to represent division between elements of two vectors $y/x$?

Question

Let $x,y \in \mathbb{R}^n$

I want to represent a vector using alternative notations:

The vector is:

$$v = \begin{bmatrix} y_1/x_1 \\ y_2/x_2 \\ \vdots \\ y_n/x_n \end{bmatrix}$$

Now one way to represent this is using:

$$\begin{bmatrix} y_1/x_1 \\ y_2/x_2 \\ \vdots \\ y_n/x_n \end{bmatrix} = yx^{-1}$$

But then you will always have to preface by defining $$x^{-1}: = \begin{bmatrix} 1/x_1 \\ 1/x_2 \\ \vdots \\ 1/x_n \end{bmatrix} $$

Is there some nicer way to represent $v$? I am thinking in line of Kronecker product, Hadamard product, or some sort of outer product

Answer 1

$$\mbox{diag} (\mathrm x) \, \mathrm v = \mathrm y$$

Assuming that $x_i \neq 0$ for all $i \in [n]$,

$$\mathrm v = \mbox{diag}^{-1} (\mathrm x) \, \mathrm y$$