Neumann series

If $A$ is a square matrix, , then $I-A$ is nonsingular and ${(I-A)}^{-1}=I+A+A^2+\mathrm{\cdots }={\sum }_{k=0}^{\mathrm{\infty }}{A}^k$. This is the Neumann series.
It provides approximations of ${(I-A)}^{-1}$ when $A$ has entries of small magnitude. For example, a first-order approximation is ${(I-A)}^{-1}\approx I+A$.
It is obvious that this is a generalization of the geometric series.

Title	Neumann series
Canonical name	NeumannSeries
Date of creation	2013-03-22 15:25:49
Last modified on	2013-03-22 15:25:49
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	9
Author	georgiosl (7242)
Entry type	Theorem
Classification	msc 15-00