The Newton Schulz method is well-known, and the proof of convergence is widely available on the internet. Yet the derivation of the method itself is more obscure. Here it is:
We seek the zero of 
The derivative of 
![\begin{array}{l} f'(X)[B] = -X^{-1} B X^{-1}. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bl%7D+f%27%28X%29%5BB%5D+%3D+-X%5E%7B-1%7D+B+X%5E%7B-1%7D.+%5Cend%7Barray%7D&bg=ffffff&fg=333333&s=0&c=20201002)
We can prove that 
![\begin{array}{l} f'^{-1}(X)[B] = -X B X. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bl%7D+f%27%5E%7B-1%7D%28X%29%5BB%5D+%3D+-X+B+X.+%5Cend%7Barray%7D&bg=ffffff&fg=333333&s=0&c=20201002)
To see this, notice that
![\begin{array}{l} B \\ = f'^{-1}(X)\Big[f'(X)[B]\Big] \\ = -X \Big[-X^{-1} B X^{-1}\Big] X \\ = B. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bl%7D+B+%5C%5C+%3D+f%27%5E%7B-1%7D%28X%29%5CBig%5Bf%27%28X%29%5BB%5D%5CBig%5D+%5C%5C+%3D+-X+%5CBig%5B-X%5E%7B-1%7D+B+X%5E%7B-1%7D%5CBig%5D+X+%5C%5C+%3D+B.+%5Cend%7Barray%7D&bg=ffffff&fg=333333&s=0&c=20201002)
The Newton method for root finding has at each iterate:
![\begin{array}{l} X_{t+1} \\ = X_t - f'^{-1}(X_t)\Big[f(X_t)\Big] \\ = X_t - f'^{-1}(X_t)\Big[X^{-1} - A\Big] \\ = X_t - X_t[X^{-1}_t-A] X_t \\ = X_t - [-X_t + X_t A X_t] \\ = 2 X_t - X_t A X_t \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bl%7D+X_%7Bt%2B1%7D+%5C%5C+%3D+X_t+-+f%27%5E%7B-1%7D%28X_t%29%5CBig%5Bf%28X_t%29%5CBig%5D+%5C%5C+%3D+X_t+-+f%27%5E%7B-1%7D%28X_t%29%5CBig%5BX%5E%7B-1%7D+-+A%5CBig%5D+%5C%5C+%3D+X_t+-+X_t%5BX%5E%7B-1%7D_t-A%5D+X_t+%5C%5C+%3D+X_t+-+%5B-X_t+%2B+X_t+A+X_t%5D+%5C%5C+%3D+2+X_t+-+X_t+A+X_t+%5Cend%7Barray%7D&bg=ffffff&fg=333333&s=0&c=20201002)
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