An example for an ambiguous Jordan-Normalform

Question

Give an example where the knowledge of the characteristic and minimal polynomial, as well as the geometric multiplicity, is not sufficient to fully determine the Jordan Canonical Form.

I'm struggling to come up with an example for this question. I know that the characteristic polynomial $p$ and the minimal polynomial $m$ alone can't determine the Jordan Canonical Form if the dimension is higher than three but every example that I've looked at becomes really simple once you know the geometric multiplicity, unless I've made some big mistake.

Answer 1

Consider a 7×7 nilpotent matrix whose characteristic polynomial is $^7$ and minimal polynomial is $^3$ with Geometric multiplicity 3. Then the Jordan form is not unique. It could be 3+3+1 or 3+2+2.

Answer 2

Hint: For a matrix in Jordan form with exactly one eigenvalue:

• The degree of the characteristic polynomial is the sum of the sizes of the Jordan blocks (i.e. the size of the matrix),
• The degree of the minimal polynomial is the size of the largest Jordan block,
• The geometric multiplicity of the eigenvalues is the number of Jordan blocks.

Further Hint: A suitable counterexample must have at least one block of size $3$ or greater.