In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a symmetric matrix and the outer product, , of vector with itself.
Statement Let denote the eigenvalues of and denote the eigenvalues of the updated matrix . In the special case when is diagonal, the eigenvectors of can be written
where is a number that makes the vector normalized.
Derivation This formula can be derived from the Sherman–Morrison formula by examining the poles of .
Remarks
The eigenvalues of were studied by Golub.
Numerical stability of the computation is studied by Gu and Eisenstat.
See also Sherman–Morrison formula
References
External links Rank-One Modification of the Symmetric Eigenproblem at EUDML Some Modified Matrix Eigenvalue Problems A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem