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.
Example
Let be given by:
We need to show that C is smaller than unity in some
norm. Therefore, we calculate:
Thus, we know from the statement above that exists.
Approximate matrix inversion
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:
The set of invertible operators is open
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
Taking the norms, we get
The norm of can be bounded by
Applications
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.