Scaled dot-product attention ($\mathrm{softmax}(Q{K}^T/\sqrt{d_k})V$)

With query and key dimension $d_k$, form the score matrix $Q{K}^T/\sqrt{d_k}$, apply the softmax function row by row, and multiply by the values $V$. If query and key coordinates have independent unit variance, their dot product has variance $d_k$; division by $\sqrt{d_k}$ keeps that score scale of order one. Optional additive masks exclude forbidden keys before the softmax function.