If \(A\) is a complex \(5 \times 5\) matrix with characteristic polynomial \[ f = (x - 2)^3(x + 7)^2 \] and minimal polynomial \(p = (x - 2)^2(x + 7)\), what is the Jordan form for \(A\)?
Solution
For the eigenvalue \(2\), the characteristic polynomial gives total multiplicity \(3\), while the exponent \(2\) in the minimal polynomial says the largest Jordan block has size \(2\). Hence the block sizes are \(2+1\).
For the eigenvalue \(-7\), the total multiplicity is \(2\), while the exponent \(1\) in the minimal polynomial says every Jordan block has size \(1\). Hence the block sizes are \(1+1\) [@HK71].
Therefore, up to permutation of Jordan blocks, \[ \boxed{ J(A)=J_2(2)\oplus J_1(2)\oplus J_1(-7)\oplus J_1(-7).} \] Explicitly, one possible ordering is \[ \begin{pmatrix} 2&1&0&0&0\\ 0&2&0&0&0\\ 0&0&2&0&0\\ 0&0&0&-7&0\\ 0&0&0&0&-7 \end{pmatrix}. \]