# 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^{}.

## References

- 1 Carl D. Meyer, Matrix Analysis and Applied Linear Algebra.

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 |