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

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

• Why don't you just call it $y/x$? – Théophile Oct 31 '16 at 2:55
• Why not just define $x^{-1}$ as the diagonal matrix with diagonal entries $1/x_i$ and write the product as $x^{-1}y$? – John Wayland Bales Oct 31 '16 at 3:34
• You have to assume further that $x_i\neq 0\ \forall i$ – P Vanchinathan Oct 31 '16 at 3:42

$$\mbox{diag} (\mathrm x) \, \mathrm v = \mathrm y$$
Assuming that $x_i \neq 0$ for all $i \in [n]$,
$$\mathrm v = \mbox{diag}^{-1} (\mathrm x) \, \mathrm y$$