A Neumann series is a mathematical series that sums k-times repeated applications of an operator
This has the generator form where
is the k-times repeated application of
This is a special case of the generalization of a geometric series of real or complex numbers to a geometric series of operators.
The generalized initial term of the series is the identity operator
and the generalized common ratio of the series is the operator
The series is named after the mathematician Carl Neumann, who used it in 1877 in the context of potential theory.
The Neumann series is used in functional analysis.
It is closely connected to the resolvent formalism for studying the spectrum of bounded operators and, applied from the left to a function, it forms the Liouville-Neumann series that formally solves Fredholm integral equations.
is a bounded linear operator on the normed vector space
If the Neumann series converges in the operator norm, then
is invertible and its inverse is the series: where
To see why, consider the partial sums Then we have This result on operators is analogous to geometric series in
One case in which convergence is guaranteed is when
in the operator norm; another compatible case is that
However, there are also results which give weaker conditions under which the series converges.
goes to infinity, the matrix norm of
This norm is confirming that the Neumann series converges.
A truncated Neumann series can be used for approximate matrix inversion.
To approximate the inverse of an invertible matrix
{\displaystyle {\begin{aligned}A^{-1}&=(I-I+A)^{-1}\\&=(I-(I-A))^{-1}\\&=(I-T)^{-1}\end{aligned}}}
Then, using the Neumann series identity that
Since these terms shrink with increasing
given the conditions on the norm, then truncating the series at some finite
may give a practical approximation to the inverse matrix: A corollary is that the set of invertible operators between two Banach spaces
is open in the topology induced by the operator norm.
, the Neumann series
can be bounded by 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 expressions for transfer functions.