# Eigengap

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{{Short description|Difference between successive eigenvalues}}
In [linear algebra](/source/linear_algebra), the '''eigengap''' of a [linear operator](/source/linear_operator) is the difference between two successive [eigenvalues](/source/eigenvalues), where eigenvalues are sorted in ascending order.

The [Davis–Kahan theorem](/source/Davis%E2%80%93Kahan_theorem), named after [Chandler Davis](/source/Chandler_Davis) and [William Kahan](/source/William_Kahan), uses the eigengap to show how eigenspaces of an operator change under [perturbation](/source/Perturbation_theory).<ref>{{cite journal |last=Davis |first=C. |author2=W. M. Kahan|date=March 1970 |title=The rotation of eigenvectors by a perturbation. III. |journal=SIAM J. Numer. Anal. |volume=7 |issue=1 |pages=1&ndash;46 |doi=10.1137/0707001 |bibcode=1970SJNA....7....1D }}</ref> Essentially, a larger eigengap indicates that the corresponding eigenvectors are more robust and less sensitive to small changes or noise in the data.

In [spectral clustering](/source/spectral_clustering), the eigengap is often referred to as the '''[spectral gap](/source/spectral_gap)'''; although the spectral gap may often be defined in a broader sense than that of the eigengap. In this context, a significant jump or gap between successive eigenvalues often suggests a natural division in the underlying data structure, helping to determine the ideal number of clusters.

== See also ==
*[Eigenvalue perturbation](/source/Eigenvalue_perturbation)
*[Spectral gap](/source/Spectral_gap)

== References ==
{{reflist}}

{{Linear-algebra-stub}}

Category:Linear algebra

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