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 matric ...
, 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 Map (mathematics), mapping V \to W between two vect ...
is the difference between two successive
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
, where eigenvalues are sorted in ascending order.
The Davis–Kahan theorem, named after
Chandler Davis
Horace Chandler Davis (August 12, 1926 – September 24, 2022) was an American-Canadian mathematician, writer, educator, and political activist: "an internationally esteemed mathematician, a minor science fiction writer of note, and among the mos ...
and
William Kahan
William "Velvel" Morton Kahan (born June 5, 1933) is a Canadian mathematician and computer scientist, who received the Turing Award in 1989 for "''his fundamental contributions to numerical analysis''",
was named an ACM Fellow in 1994, and induc ...
, uses the eigengap to show how eigenspaces of an operator change under
perturbation
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbatio ...
.
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 a ...
, 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 eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a sc ...
''; although the spectral gap may often be defined in a broader sense than that of the eigengap.
See also
*
Eigenvalue perturbation
In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system Ax=\lambda x that is perturbed from one with known eigenvectors and eigenvalues A_0 x=\lambda_0x_0 . This is useful for studyi ...
References
{{Linear-algebra-stub
Linear algebra