# inverse of matrix with small-rank adjustment

Suppose that an $n\times n$ matrix $B$ is obtained by adding a small-rank adjustment $XR{Y}^{T}$ to matrix $A$,

$$B=A+XR{Y}^{T},$$ |

where $X$ and $Y$ are $n\times r$ matrices, and $R$ is an $r\times r$ matrix. Assume that the inverse of $A$ is known and $r$ is much smaller than $n$. The following formula for ${B}^{-1}$ is often useful,

$${B}^{-1}={A}^{-1}-{A}^{-1}X{({R}^{-1}+{Y}^{T}{A}^{-1}X)}^{-1}{Y}^{T}{A}^{-1}$$ |

provided that all inverses in the formula exist.

In particular, when $r=1$ and $A=I$, we have

$${(I+x{y}^{T})}^{-1}=I-\frac{x{y}^{T}}{1+{y}^{T}x}$$ |

for any $n\times 1$ column vectors^{} $x$ and $y$ such that $1+{y}^{T}x\ne 0$.

Title | inverse of matrix with small-rank adjustment |
---|---|

Canonical name | InverseOfMatrixWithSmallrankAdjustment |

Date of creation | 2013-03-22 15:46:06 |

Last modified on | 2013-03-22 15:46:06 |

Owner | kshum (5987) |

Last modified by | kshum (5987) |

Numerical id | 8 |

Author | kshum (5987) |

Entry type | Theorem |

Classification | msc 15A09 |