Editing Canonical basis (section)

===Example===
This example illustrates a canonical basis with two Jordan chains.  Unfortunately, it is a little difficult to construct an interesting example of low order.<ref>{{harvtxt|Nering|1970|pp=122,123}}</ref>
The matrix

:<math>A = \begin{pmatrix} 
4 & 1 & 1 & 0 & 0 & -1 \\
0 & 4 & 2 & 0 & 0 & 1 \\
0 & 0 & 4 & 1 & 0 & 0 \\
0 & 0 & 0 & 5 & 1 & 0 \\
0 & 0 & 0 & 0 & 5 & 2 \\
0 & 0 & 0 & 0 & 0 & 4
\end{pmatrix}</math>

has eigenvalues <math> \lambda_1 = 4 </math> and <math> \lambda_2 = 5 </math> with algebraic multiplicities <math> \mu_1 = 4 </math> and <math> \mu_2 = 2 </math>, but [[geometric multiplicity|geometric multiplicities]] <math> \gamma_1 = 1 </math> and <math> \gamma_2 = 1 </math>.

For <math> \lambda_1 = 4,</math> we have <math> n - \mu_1 = 6 - 4 = 2, </math>

:<math> (A - 4I) </math> has rank 5,
:<math> (A - 4I)^2 </math> has rank 4,
:<math> (A - 4I)^3 </math> has rank 3,
:<math> (A - 4I)^4 </math> has rank 2.

Therefore <math>m_1 = 4.</math>

:<math> \rho_4 = \operatorname{rank}(A - 4I)^3 - \operatorname{rank}(A - 4I)^4 = 3 - 2 = 1,</math>
:<math> \rho_3 = \operatorname{rank}(A - 4I)^2 - \operatorname{rank}(A - 4I)^3 = 4 - 3 = 1,</math>
:<math> \rho_2 = \operatorname{rank}(A - 4I)^1 - \operatorname{rank}(A - 4I)^2 = 5 - 4 = 1,</math>
:<math> \rho_1 = \operatorname{rank}(A - 4I)^0 - \operatorname{rank}(A - 4I)^1 = 6 - 5 = 1.</math>

Thus, a canonical basis for <math>A</math> will have, corresponding to <math> \lambda_1 = 4,</math> one generalized eigenvector each of ranks 4, 3, 2 and 1.

For <math> \lambda_2 = 5,</math> we have <math> n - \mu_2 = 6 - 2 = 4, </math>

:<math> (A - 5I) </math> has rank 5,
:<math> (A - 5I)^2 </math> has rank 4.

Therefore <math>m_2 = 2.</math>

:<math> \rho_2 = \operatorname{rank}(A - 5I)^1 - \operatorname{rank}(A - 5I)^2 = 5 - 4 = 1,</math>
:<math> \rho_1 = \operatorname{rank}(A - 5I)^0 - \operatorname{rank}(A - 5I)^1 = 6 - 5 = 1.</math>

Thus, a canonical basis for <math>A</math> will have, corresponding to <math> \lambda_2 = 5,</math> one generalized eigenvector each of ranks 2 and 1.

A canonical basis for <math>A</math> is

:<math>
\left\{ \mathbf x_1, \mathbf x_2, \mathbf x_3, \mathbf x_4, \mathbf y_1, \mathbf y_2 \right\} =
\left\{
\begin{pmatrix} -4 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix},
\begin{pmatrix} -27 \\ -4 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix},
\begin{pmatrix} 25 \\ -25 \\ -2 \\ 0 \\ 0 \\ 0 \end{pmatrix},
\begin{pmatrix} 0 \\ 36 \\ -12 \\ -2 \\ 2 \\ -1 \end{pmatrix},
\begin{pmatrix} 3 \\ 2 \\ 1 \\ 1 \\ 0 \\ 0 \end{pmatrix},
\begin{pmatrix} -8 \\ -4 \\ -1 \\ 0 \\ 1 \\ 0 \end{pmatrix} 
\right\}.
</math>

<math> \mathbf x_1 </math> is the ordinary eigenvector associated with <math> \lambda_1 </math>.  
<math> \mathbf x_2, \mathbf x_3 </math> and <math> \mathbf x_4 </math> are generalized eigenvectors associated with <math> \lambda_1 </math>.  
<math> \mathbf y_1 </math> is the ordinary eigenvector associated with <math> \lambda_2 </math>.  
<math> \mathbf y_2 </math> is a generalized eigenvector associated with <math> \lambda_2 </math>.

A matrix <math>J</math> in Jordan normal form, similar to <math>A</math> is obtained as follows:

:<math>
M =
\begin{pmatrix} \mathbf x_1 & \mathbf x_2 & \mathbf x_3 & \mathbf x_4 & \mathbf y_1 & \mathbf y_2 \end{pmatrix} =
\begin{pmatrix}
-4 & -27 &  25 &   0 & 3 & -8 \\
 0 &  -4 & -25 &  36 & 2 & -4 \\
 0 &   0 &  -2 & -12 & 1 & -1 \\
 0 &   0 &   0 &  -2 & 1 &  0 \\
 0 &   0 &   0 &   2 & 0 &  1 \\
 0 &   0 &   0 &  -1 & 0 &  0
\end{pmatrix},
</math>
:<math> J = \begin{pmatrix}
4 & 1 & 0 & 0 & 0 & 0 \\
0 & 4 & 1 & 0 & 0 & 0 \\
0 & 0 & 4 & 1 & 0 & 0 \\
0 & 0 & 0 & 4 & 0 & 0 \\
0 & 0 & 0 & 0 & 5 & 1 \\
0 & 0 & 0 & 0 & 0 & 5
\end{pmatrix},
</math>

where the matrix <math>M</math> is a [[generalized modal matrix]] for <math>A</math> and <math>AM = MJ</math>.<ref>{{harvtxt|Bronson|1970|p=203}}</ref>