Bunch–Nielsen–Sorensen formula - Alchetron, the free social encyclopedia

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

Statement

Let λ i denote the eigenvalues of A and λ ~ i denote the eigenvalues of the updated matrix A ~ = A + v v T . In the special case when A is diagonal, the eigenvectors q ~ i of A ~ can be written

( q ~ i ) k = N i v k λ k − λ ~ i

where N i is a number that makes the vector q ~ i normalized.

Derivation

This formula can be derived from the Sherman–Morrison formula by examining the poles of ( A − λ ~ + v v T ) − 1 .

Remarks

The eigenvalues of A ~ were studied by Golub.

Numerical stability of the computation is studied by Gu and Eisenstadt.

References

Bunch–Nielsen–Sorensen formula Wikipedia

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Contents

Statement

Derivation

Remarks

References