In
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 matrix (mathemat ...
, the eigengap of a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
is the difference between two successive
eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
, where eigenvalues are sorted in ascending order.
The Davis–Kahan theorem, named after
Chandler Davis and
William Kahan
William "Velvel" Morton Kahan (born June 5, 1933) is a Canadian mathematician and computer scientist, who is a professor emeritus at University of California, Berkeley. He received the Turing Award in 1989 for "his fundamental contributions to nu ...
, uses the eigengap to show how eigenspaces of an operator change under
perturbation.
In
spectral clustering
In multivariate statistics, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided ...
, the eigengap is often referred to as the ''
spectral gap
In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to oth ...
''; although the spectral gap may often be defined in a broader sense than that of the eigengap.
See also
*
Eigenvalue perturbation
References
{{Linear-algebra-stub
Linear algebra