Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If it is not true, can you provide a counter-example?

share|improve this question
    
Really? How about {{1,1},{0,1}} and {{1,0},{0,1}}? They are dissimilar but they have the same characteristic polynomial. –  Voldemort Sep 19 '12 at 1:47
3  
@anon, that's wrong. Not even having the same characteristic and minimal polynomial is enough. –  DonAntonio Sep 19 '12 at 1:57
1  
Not sure what I was thinking there, hmm. –  anon Sep 19 '12 at 2:47
    

2 Answers 2

up vote 7 down vote accepted

Consider matrices in Jordan normal form with the same diagonal entries. The minimal polynomial just tells you the size of the biggest Jordan blocks (for the respective Eigenvalues). Example (for some $a$):

$\begin{pmatrix} a & 1 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a\end{pmatrix},\begin{pmatrix} a & 1 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 1 \\ 0 & 0 & 0 & a\end{pmatrix}$

share|improve this answer

Take the matrices

$$\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{pmatrix}\,\,,\,\,\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}$$

These two matrices have $\,x^4\,$ as char. polynomial and $\,x^2\,$ as minimal one.

Try a nice exercise: prove that the condition is sufficient if the matrix is $\,n\times n\,\,,\,n\leq 3\,$

share|improve this answer
    
Insufficient in general though... $n>4$. –  Squirtle Dec 4 '13 at 23:00
    
What about this... [0 0;0 0] and [0 1;0 0] have the same char poly but they aren't similar –  Squirtle Dec 4 '13 at 23:42
    
Read the question carefully:same characteristic and minimal polynomials –  DonAntonio Dec 5 '13 at 0:25
    
Oooooooooooooops –  Squirtle Dec 5 '13 at 0:25

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.