In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in particular
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, 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
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with re ...
and the
outer product
In linear algebra, the outer product of two coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An ea ...
,
, of
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
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
In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix A and the outer product, u v^\textsf, of vectors u and v. The ...
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
In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix A and the outer product, u v^\textsf, of vectors u and v. The ...
References
External links
Rank-One Modification of the Symmetric Eigenproblema
EUDMLSome Modified Matrix Eigenvalue ProblemsA Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem
{{DEFAULTSORT:Bunch-Nielsen-Sorensen formula
Linear algebra