A Neumann series is a
mathematical series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
of the form
:
where
is an
operator and
its
times repeated application.
This generalizes the
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...
.
The series is named after the mathematician
Carl Neumann
Carl Gottfried Neumann (also Karl; 7 May 1832 – 27 March 1925) was a German mathematician.
Biography
Neumann was born in Königsberg, Prussia, as the son of the mineralogist, physicist and mathematician Franz Ernst Neumann (1798–1895), who w ...
, who used it in 1877 in the context of
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
. The Neumann series is used in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
. It forms the basis of the
Liouville-Neumann series, which is used to solve
Fredholm integral equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to so ...
s. It is also important when studying the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of bounded operators.
Properties
Suppose that
is a bounded linear operator on the
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
. If the Neumann series
converges in the
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introd ...
, then
is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
and its inverse is the series:
:
,
where
is the
identity operator
Identity may refer to:
* Identity document
* Identity (philosophy)
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* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film) ...
in
. To see why, consider the partial sums
:
.
Then we have
:
This result on operators is analogous to
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...
in
, in which we find that:
:
:
One case in which convergence is guaranteed is when
is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
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.
Another application is the theory of
Propagation graphs which takes advantage of Neumann series to derive closed form expression for the transfer function.
References
* {{cite book, last=Werner, first=Dirk, year=2005, title=Funktionalanalysis , language=de, publisher=Springer Verlag, isbn=3-540-43586-7
Functional analysis
Mathematical series