1

This question comes out of the conversation in the comments of this answer. The answerer asserts the following: $\DeclareMathOperator{\rank}{rank}$

Suppose that $A$ and $B$ have the same minimal polynomial and that for all $\lambda \in \Bbb C$, $\rank(A - \lambda I) = \rank(B - \lambda I)$. Then $A$ and $B$ are similar.

My question: is this true or false? I think it's false, and will attempt to build a counterexample as an answer. However, I welcome any attempts in either direction.

Community
  • 1
Ben Grossmann
  • 234,171
  • 12
  • 184
  • 355

1 Answers1

2

$\DeclareMathOperator{\rank}{rank}$The answer is no. Notably, the minimal polynomial determines the size of the largest blocks in the Jordan form, $\rank(A - \lambda I)$ determines the total number of blocks.

Let $J_k$ denote the $\lambda = 0$ block of size $k$. Consider the matrices $$ A = J_3 \oplus J_2 \oplus J_2\\ B = J_3 \oplus J_3 \oplus J_1 $$ $A$ and $B$ have the same characteristic polynomial and minimal polynomial. They each have only the eigenvalue $\lambda = 0$, and satisfy $\rank(A) = \rank(B)$. However, they are not similar.

Martin Sleziak
  • 56,060
Ben Grossmann
  • 234,171
  • 12
  • 184
  • 355
  • @EricTowers Not so. Both matrices have rank $7-3 = 4$, since there are $3$ Jordan blocks associated with $0$. To put it another way: each of these matrices has $3$ zero-columns, and the other columns are clearly linearly independent. – Ben Grossmann Nov 13 '16 at 20:27
  • @copper.hat need to wait two days, but thanks for the endorsement – Ben Grossmann Nov 13 '16 at 21:38