Rank Modifications of Semidefinite Matrices Associated with a Secant Update Formula
Moody T. Chu, Robert E. Funderlic, Gene Howard Golub · SIAM Journal on Matrix Analysis and Applications · 1998
This paper analyzes rank modification of symmetric positive definite matrices H of the form H- M+ P, where H- M denotes a step of reducing H to a lower-rank, symmetric and positive semidefinite matrix and ( H- M)+ P denotes a step of restoring H- M to a symmetric positive definite matrix. These steps and their generalizations for rectangular matrices are fully characterized. The well-known BFGS and DFP updates used in Hessian and inverse Hessian approximations provided the motivation for the generalizations and are special cases with H and P having rank one.