Rank-one and Rank-two Corrections to Positive Definite Matrices Expressed in Product Form

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Abstract
It is shown that certain rank-one and rank-two corrections to symmetric positive definite matrices may be expressed in the form of a product. This product form gives control over the positive definiteness, determinant value and conditioning of the corrected matrix. An application to updating formulae of quasi-Newton methods for unconstrained minimization is discussed.