inverse of matrix with small-rank adjustment

Suppose that an n×n matrix B is obtained by adding a small-rank adjustment XRYT to matrix A,


where X and Y are n×r matrices, and R is an r×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,


provided that all inverses in the formula exist.

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


for any n×1 column vectorsMathworldPlanetmath x and y such that 1+yTx0.

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