Find all possible Jordan canonical forms of $4\times 4$ complex matrices which satisfy the following three conditions simultaneously

Question

Find all possible Jordan canonical forms of $4\times 4$ complex matrices which satisfy the following three conditions simultaneously：

1. $A$ is not diagonalizable;

2. $A$ has characteristic polynomial $(x-5)^2(x-1)^2$;

3. $A$ satisfies the equation $(A-5I)(A-I)^3=0.$

Answer 1

Hint:

• From 1, deduce that at least one block has size $2$ or larger,
• From 2, deduce that both $5$ and $1$ are eigenvalues,
• From 3, deduce that the largest block associated with $\lambda = 5$ has size $1$.