Editing Invertible matrix (section)

=== By Neumann series ===
If a matrix {{math|'''A'''}} has the property that
: <math>\lim_{n \to \infty} (\mathbf I - \mathbf A)^n = 0</math>

then {{math|'''A'''}} is nonsingular and its inverse may be expressed by a [[Neumann series]]:<ref>
{{cite book
  | last = Stewart
  | first = Gilbert
  | title = Matrix Algorithms: Basic decompositions
  | publisher = SIAM
  | year = 1998
  | pages = 55
  | isbn = 978-0-89871-414-2
}}</ref>

: <math>\mathbf A^{-1} = \sum_{n = 0}^\infty (\mathbf I - \mathbf A)^n.</math>

Truncating the sum results in an "approximate" inverse which may be useful as a [[preconditioner]]. Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a [[geometric sum]]. As such, it satisfies 
: <math>\sum_{n=0}^{2^L-1} (\mathbf I - \mathbf A)^n = \prod_{l=0}^{L-1}\left(\mathbf I + (\mathbf I - \mathbf A)^{2^l}\right)</math>.

Therefore, only {{math|2''L'' − 2}} matrix multiplications are needed to compute {{math|2{{sup|''L''}}}} terms of the sum.

More generally, if {{math|'''A'''}} is "near" the invertible matrix {{math|'''X'''}} in the sense that
: <math>\lim_{n \to \infty} \left(\mathbf I - \mathbf X^{-1} \mathbf A\right)^n = 0 \mathrm{~~or~~} \lim_{n \to \infty} \left(\mathbf I - \mathbf A \mathbf X^{-1}\right)^n = 0</math>

then {{math|'''A'''}} is nonsingular and its inverse is
: <math>\mathbf A^{-1} = \sum_{n = 0}^\infty \left(\mathbf X^{-1} (\mathbf X - \mathbf A)\right)^n \mathbf X^{-1}~.</math>

If it is also the case that {{math|'''A''' − '''X'''}} has [[rank (linear algebra)|rank]] 1 then this simplifies to
: <math>\mathbf A^{-1} = \mathbf X^{-1} - \frac{\mathbf X^{-1} (\mathbf A - \mathbf X) \mathbf X^{-1}}{1 + \operatorname{tr}\left(\mathbf X^{-1} (\mathbf A - \mathbf X)\right)}~.</math>