Problem on characteristic polynomial and minimal polynomial

Asked Oct 11 '17 at 13:39

Active Oct 11 '17 at 13:47

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I am given two matrices a and b such that characteristic polynomial and minimal polynomial of a and b are equal

I have to check

My attempt:

I know a and are not similar for size of matrix greater than 3 So they do not have same JC form also

I also conclude that as they are not similar I can't be sure that if a is diagonalisable then b must be diagonalisable

How can I give argument for 3

Am I correct?

asked Oct 11 '17 at 13:39

Abhishek Chandra

1

4 is actually true – A.Γ. Oct 11 '17 at 13:45
What about 3 ?? – Abhishek Chandra Oct 11 '17 at 13:46
1

A counterexample for 3 can be constructed for $4\times 4$ nilpotent matrices via Jordan forms in the spirit of 2: just pick the largest block to be $\begin{bmatrix}0 & 1\0 & 0\end{bmatrix}$ and see how you can fill up the rest differently. – A.Γ. Oct 11 '17 at 13:52

0 Answers0