where is the identity operator in . To see why, consider the partial sums
.
Then we have
This result on operators is analogous to geometric series in , in which we find that:
One case in which convergence is guaranteed is when is a Banach space and in the operator norm or is convergent. However, there are also results which give weaker conditions under which the series converges.
A truncated Neumann series can be used for approximate matrix inversion. To approximate the inverse of an invertible matrix , we can assign the linear operator as:
where is the identity matrix. If the norm condition on is satisfied, then truncating the series at , we get:
A corollary is that the set of invertible operators between two Banach spaces and is open in the topology induced by the operator norm. Indeed, let be an invertible operator and let be another operator.
If , then is also invertible.
Since , the Neumann series is convergent. Therefore, we have
The Neumann series has been used for linear data detection in massive multiuser multiple-input multiple-output (MIMO) wireless systems. Using a truncated Neumann series avoids computation of an explicit matrix inverse, which reduces the complexity of linear data detection from cubic to square.[1]
Another application is the theory of Propagation graphs which takes advantage of Neumann series to derive closed form expression for the transfer function.