Canonical Basis Can you show how wikipedia got the following

Canonical Basis.

Can you show how wikipedia got the following equation? Thank you!

Computation [edit] Let lambda_i be an eigenvalue of A of algebraic multiplicity mu. First, find the ranks (matrix ranks) of the matrices (A - lambda_i I), (A - lambda_i I)^2, ..., (A- lambda_i I)^m_i. The integer m_i is determined to be the first integer for which (A - lambda_iI)m_i has rank n - mu_i (n being the number of rows or columns of A, that is, A is n x n). Now define pk=rank(A-lambda_iI)^k-1 - rank(A - lambda_i I)^k (k=1, 2, ..., m_i). The variable pk designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue lambda_i that will appear in a canonical basis for A.

Solution

In mathematics sometime we first assume the statement and we prove it by examples.

This statement isalso assumed and we prove it only by examples.After this statement one example is given that proves this statement true