Give the \(4 \times 4\) Jordan forms with minimal polynomial \((x - 1)(x - 2)^2\).
Solution
The minimal polynomial \[ m_A(x)=(x-1)(x-2)^2 \] says exactly the following about the Jordan blocks:
- for eigenvalue \(1\), every Jordan block has size \(1\), and at least one such block occurs;
- for eigenvalue \(2\), every Jordan block has size at most \(2\), and at least one block has size exactly \(2\).
Since the total dimension is \(4\), enumerate the possible partitions of the generalized eigenspace dimensions subject to those constraints.
1. The eigenvalue-\(2\) generalized eigenspace has dimension \(2\).Then it must contribute one block \(J_2(2)\), while eigenvalue \(1\) contributes two \(1\times1\) blocks:\[ J_2(2)\oplus[1]\oplus[1]. \]
2. The eigenvalue-\(2\) generalized eigenspace has dimension \(3\).It must contain one block of size \(2\) and one block of size \(1\), while eigenvalue \(1\) contributes one block:\[ J_2(2)\oplus[2]\oplus[1]. \]
No other case is possible. The eigenvalue-\(2\) generalized eigenspace cannot have dimension \(4\), because the factor \((x-1)\) in the minimal polynomial requires eigenvalue \(1\) to occur; and it cannot have dimension \(1\), because \((x-2)^2\) requires a size-\(2\) Jordan block.
Hence, up to permutation of Jordan blocks, the only Jordan forms are \[ \boxed{J_2(2)\oplus[1]\oplus[1]} \qquad\text{and}\qquad \boxed{J_2(2)\oplus[2]\oplus[1]}. \]