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 matrices ...
, the eigengap of a
linear operator is the difference between two successive
eigenvalues, where eigenvalues are sorted in ascending order.
The Davis–Kahan theorem, named after
Chandler Davis and
William Kahan, 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 as ...
, 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 othe ...
''; 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