[D]In transformer models, why is there a query and key matrix instead of just the product?

The only time that the query and key matrices are used is to compute the attention scores. That is $v\_i\^T \\cdot W\_q\^T W\_k v\_j$ But what is used is the matrix $W\_q\^T W\_k$. Why not just replace $W\_q\^T W\_k$ with a single matrix $W\_{qv}$, and learn the matrix that is the product of W\_q\^T W\_k instead of the matrices themselves? How does it help to have two matrices instead of one? And if it helps, why is that not done …

attention compute machinelearning matrix product query the matrix transformer transformer models