Editing Canonical basis (section)

===Computation===
Let <math> \lambda_i </math> be an eigenvalue of <math>A</math> of algebraic multiplicity <math> \mu_i </math>.  First, find the [[rank (linear algebra)|ranks]] (matrix ranks) of the matrices <math> (A - \lambda_i I), (A - \lambda_i I)^2, \ldots , (A - \lambda_i I)^{m_i} </math>.  The integer <math>m_i</math> is determined to be the ''first integer'' for which <math> (A - \lambda_i I)^{m_i} </math> has rank <math>n - \mu_i </math> (''n'' being the number of rows or columns of <math>A</math>, that is, <math>A</math> is ''n'' × ''n'').

Now define

:<math> \rho_k = \operatorname{rank}(A - \lambda_i I)^{k-1} - \operatorname{rank}(A - \lambda_i I)^k \qquad (k = 1, 2, \ldots , m_i).</math>

The variable <math> \rho_k </math> designates the number of linearly independent generalized eigenvectors of rank ''k'' (generalized eigenvector rank; see [[generalized eigenvector#Overview and definition|generalized eigenvector]]) corresponding to the eigenvalue <math> \lambda_i </math> that will appear in a canonical basis for <math>A</math>.  Note that

:<math> \operatorname{rank}(A - \lambda_i I)^0 = \operatorname{rank}(I) = n .</math>

Once we have determined the number of generalized eigenvectors of each rank that a canonical basis has, we can obtain the vectors explicitly (see [[Generalized eigenvector#Computation of generalized eigenvectors|generalized eigenvector]]).<ref>{{harvtxt|Bronson|1970|pp=197,198}}</ref>