Jordan Canonical Form of a Matrix Given Characteristic and Minimum Polynomials

If we have the characteristic polynomial  
\[f( \lambda )\]
  and minimum polynomial  
\[m( \lambda )\]
  of a matrix  
\[A\]
  then we can find the Jordan canonical form of  
\[A\]

Suppose  
\[m( \lambda)= (2- \lambda )^4(5- \lambda), \: f( \lambda)= (2- \lambda )^5(5- \lambda)^3 \]

The Jordan canonical form of the matrix is then
\[ \left( \begin{array}{cccccccc} 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 \end{array} \right) \]
.

Add comment

Security code
Refresh