Silva Almeida's Rule

W+=W+α(W−[Y][Z]T)=W+α(W−[WX][WT[WX]]T)=W+α(W−WXXTWTW)=W+α(Id−WXXTWT)W=W+α(Id−(WX)(WX)T)W=W+α(Id−YYT)W\begin{aligned} W_{+} & =W+\alpha\left(W-[Y][Z]^T\right) \\ & =W+\alpha\left(W-[W X]\left[W^T[W X]\right]^T\right) \\ & =W+\alpha\left(W-W X X^T W^T W\right) \\ & =W+\alpha\left(I d-W X X^T W^T\right) W \\ & =W+\alpha\left(I d-(W X)(W X)^T\right) W \\ & =W+\alpha\left(I d-Y Y^T\right) W \end{aligned}