How many possible Jordan forms are there for a \(6 \times 6\) complex matrix with characteristic polynomial \((x + 2)^4(x - 1)^2\)?
Solution
Jordan forms are determined, for each eigenvalue separately, by a partition of its algebraic multiplicity into Jordan block sizes [@HK71].
For eigenvalue \(-2\), multiplicity \(4\), the five partitions are \[ 4,\qquad 3+1,\qquad 2+2,\qquad 2+1+1,\qquad 1+1+1+1. \] For eigenvalue \(1\), multiplicity \(2\), the two partitions are \[ 2,\qquad 1+1. \] The choices are independent. Therefore the number of Jordan forms, up to permutation of blocks, is \[ \boxed{5\cdot2=10}. \]