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: