whichafter similar calculation as the one aboveyields ##in onxk ( bd- )- onxk in -bd- - ik - - ik okxn - okxn ( the upper-left block of - from ( must be the same as the upper-left block of - from ( )leading to the woodbury identity( bd- )- - - ( ca- )- ca- ( det( bd- cdet(ddet(adet( ca- ( woodbury identity from ( and the fact that the determinant of product is the product of the determinantswe see that det(sdet(adet( ca- bsimilarlyfrom ( we have det(sdet( bd- cdet( )leading to the identity the following special cases of ( and ( are of particular importance theorem sherman-morrison formula suppose that rnxn is invertible and xy rn thendet( xydet( )( ya- xif in addition ya- - then the sherman-morrison formula holds( xy)- - shermanmorrison formula - xya- ya- prooftake xc -yand in ( and ( one important application of the sherman-morrison formula is in the efficient solution of the linear system ax bwhere is an matrix of the forma > = for some column vectors rn and diagonal (or otherwise easily invertiblematrix such linear systems arisefor examplein the context of ridge regression and optimization to see how the sherman-morrison formula can be exploiteddefine the matrices via the recursionak ak- ak > application of theorem for yields the identities: - - ak- - - - ak ak ak- > - - ak |ak |ak- > - - ak here |ais shorthand notation for det(