\[A\]
has minimum polynomial \[m(\lambda)=(2- \lambda)^2\]
.We wish to find the Jordan canonical form of
\[A\]
.The order of the characteristic polynomial is the same order as the matrix, and since the minimum polynomial includes all roots of the characteristic polynomial, we must have
\[m(\lambda)=(2- \lambda)^5\]
.The order of Jordan Blocks in the Jordan canonical form of
\[A\]
is two, equal to the order of the minimum polynomial. We can fit at most two Jordan block matrices on the diagonal of the Jordan Canonical Form. This must be at the top left of the main diagonal, being the largest blocks.The Jordan Canonical Form can only be
\[ \left( \begin{array}{ccccc} 2 & 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{array} \right) \]
or \[ \left( \begin{array}{ccccc} 2 & 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{array} \right) \]