Bunch–Nielsen–Sorensen formula explained

A and the outer product, v v ^{T} , of vector v with itself.

Statement

Let

λ _{i} denote the eigenvalues of A and \tilde λ _{i} denote the eigenvalues of the updated matrix \tilde A = A + v v ^{T} . In the special case when A is diagonal, the eigenvectors \tilde q _{i} of \tilde A can be written(\tilde q _{i)} _{k} = (N _{i} v _{k})/(λ _{k} - \tilde λ _{i})

where

N _{i} is a number that makes the vector \tilde q _{i} normalized.

Derivation

This formula can be derived from the Sherman–Morrison formula by examining the poles of

(A - \tilde λ I + vv ^{T)} ^{-1} .

Remarks

The eigenvalues of

\tilde A were studied by Golub.^[2]

Numerical stability of the computation is studied by Gu and Eisenstat.^[3]

See also

• Sherman–Morrison formula

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

Notes and References

1. Bunch . J. R. . Nielsen . C. P. . Sorensen . D. C. . Rank-one modification of the symmetric eigenproblem . 10.1007/BF01396012 . Numerische Mathematik . 31 . 31–48 . 1978 . 120776348 .
2. Golub . G. H. . Some Modified Matrix Eigenvalue Problems . 10.1137/1015032 . SIAM Review . 15 . 2 . 318–334 . 1973 . 10.1.1.454.9868 .
3. Gu . M. . Eisenstat . S. C. . 10.1137/S089547989223924X . A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem . SIAM Journal on Matrix Analysis and Applications . 15 . 4 . 1266 . 1994 .