Can I cut matrix $W$ in the fraction $\frac{W}{WH}$ when $WH$ is matrix multiplication?

Question

$W_{ik}$ is matrix with $i$ rows and $k$ columns, $H_{kj}$ has $k$ rows and $j$ columns. $\frac{W_{ik}}{W_{ik}H_{kj}}$ is the element-wise division of matrix but $W_{ik}H_{kj}$ is matrix multiplication. In this case, can $W_{ik}$ be cut out from the fraction? Why?

Answer 1

If you assume all the stars align so that all parts of this are defined, the answer is still no. Let $W$ be the $2 \times 2$ identity matrix and let $H= \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$. Then $WH=H$ and $\frac{W}{WH}= W$, the identity matrix. You want this to be $\frac{1}{H}$ which I can only assume you wish to interpret as $H^{-1}$. However, $H^{-1}$ is not the identity matrix. Even worse, $H^{-1}$ doesn't even exist.

If, instead, you wish to interpret cancellation as $\frac{W}{WH}= \frac{I}{H}$ then $W = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}$ and $H$ as above give a counterexample.