An '''eigenbasis''' for a linear operator <math>T</math> that operates on a vector space <math>V</math> is a basis for <math>V</math> that consists entirely of eigenvectors of <math>T</math> (possibly with different eigenvalues). Such a basis mayexists notprecisely existif the direct sum of the eigenspaces equals the whole space, in which case one can take the union of bases chosen in each of the eigenspaces as eigenbasis. The matrix of ''T'' in a given basis is diagonal precisely when that basis is an eigenbasis for ''T'', and for this reason ''T'' is called '''diagonalizable''' if it admits an eigenbasis.
Suppose <math>V</math> has finite dimension <math>n</math>, and let <math>\boldsymbol{\gamma}_T</math> be the sum of the geometric multiplicities <math>\gamma_T(\lambda_i)</math> over all distinct eigenvalues <math>\lambda_i</math> of <math>T</math>. This integer is the maximum number of linearly independent eigenvectors of <math>T</math>, and therefore cannot exceed <math>n</math>. If <math>\boldsymbol{\gamma}_T</math> is exactly <math>n</math>, then <math>T</math> admits an eigenbasis; that is, there exists a basis for <math>V</math> that consists of <math>n</math> eigenvectors. The matrix <math>A</math> that represents <math>T</math> relative to this basis is a diagonal matrix, whose diagonal elements are the eigenvalues associated to each basis vector.
Conversely, if the sum <math>\boldsymbol{\gamma}_T</math> is less than <math>n</math>, then <math>T</math> admits no eigenbasis, and there is no choice of coordinates that will allow <math>T</math> to be represented by a diagonal matrix.
Note that <math>\boldsymbol{\gamma}_T</math> is at least equal to the number of ''distinct'' eigenvalues of <math>T</math>, but may be larger than that.<ref>{{Citation|first=Gilbert|last=Strang|title=Linear Algebra and Its Applications|edition=3rd|publisher=Harcourt|___location= San Diego| year=1988}}</ref> For example, the identity operator <math>I</math> on <math>V</math> has <math>\boldsymbol{\gamma}_I = n</math>, and any basis of <math>V</math> is an eigenbasis of <math>I</math>; but its only eigenvalue is 1, with <math>\gamma_T(1) = n</math>.
==Generalizations to infinite-dimensional spaces==
