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 , defined as follows:
The derivative of at
, applied to matrix
, operates on
as follows:
We can prove that at
, applied to matrix
, operates on
as follows:
To see this, notice that
The Newton method for root finding has at each iterate: