Similar matrices and polynomials

Asked May 09 '16 at 10:54

Active May 09 '16 at 20:41

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Any two matrices M and N which have the same characteristic and minimal polynomial are similar ?

Is there any easier way to explain ?

I tried as M and N have same Jordan forms. So let Jordan form J of M implies J = P^-1MP and Jordan form K of N give K = Q^-1NQ & J=K gives after solving N = R^-1MR where R = PQ^-1. Gives M and N are similar.

edited May 09 '16 at 20:41

user26857

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asked May 09 '16 at 10:54

Shona

M and N are similar matrices ? They have same characteristic and minimal polynomials . – Shona May 09 '16 at 11:11
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If their size is $\le 3$, yes. If it's $\ge 4$, not necessarily. – May 09 '16 at 11:15
@G. Sassatelli can you pls elaborate ? – Shona May 09 '16 at 11:19
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See "Jordan Canonical Form determined by characteristic and minimal polynomials in dimension $3$, but not beyond". – May 09 '16 at 11:25
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The same procedure can be applied to matrices over any algebraically closed field. Moreover, since two matrices in $M_n(\Bbb K)$ are similar if and only if they are similar in $M_n\left(\overline{\Bbb K}\right)$, the result can be extended to arbitrary fields. – May 09 '16 at 11:25
@G. Sassatelli is there any easier way to explain it ? – Shona May 09 '16 at 11:37
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Well, technically, just the counterexample in $M_4(\Bbb C)$ (which actually works for $M_4(\Bbb K)$) solves your question. As for the remaining part, in my humble opinion, I think the use of Jordan canonical form, for the complex case, is somehow needed, because the proof I am aware of ultimately boils down to "if you classify the possible Jordan forms of a $n\times n$ complex matrix, it turns out that there are not that many of them for $n\le 3$, while for $n>3$ there are". – May 09 '16 at 12:34
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I don't know if the generalization to arbitrary fields was within your interests, but I think that, as algebraic as the details may be, the lemma on similarity that I quoted (at least in the version I used) is somehow needed to face the full problem. If you were interested only in complex matrices, you can ignore it. – May 09 '16 at 12:39
Just abit difficult for me . can you explain more easily ? – Shona May 09 '16 at 13:22
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As I said, I cannot. – May 09 '16 at 14:50
@G. Sassatelli ok thank you for you time and help ☺ – Shona May 09 '16 at 15:07

Similar matrices and polynomials

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