# Characterstic & Minimal polynomials.

• Mar 23rd 2010, 11:56 AM
Also sprach Zarathustra
Characterstic & Minimal polynomials.
Prove or disprove(by giving an example) the following statements:

1. Matrices wit the same characteristic polynomial and minimal polynomial are similar.

2. Diagonalizable matrices with the same characteristic polynomial are similar.

3. Diagonalizable matrices with the same minimal polynomial are similar.

4.
Matrices with the same characteristic polynomial that factorize to different linear factors, are(matrices) similar.

5.Matrices with the same characteristic polynomial that factorize to different irreducible polynomial, are with the same minimal polynomial.

• Mar 23rd 2010, 02:33 PM
tonio
Quote:

Originally Posted by Also sprach Zarathustra
Prove or disprove(by giving an example) the following statements:

1. Matrices wit the same characteristic polynomial and minimal polynomial are similar.

False, counterexample: $A=\begin{pmatrix}1&1&0&0\\0&1&0&0\\0&0&1&1\\0&0&0& 1\end{pmatrix}\,,\,\,B=\begin{pmatrix}1&1&0&0\\0&1 &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$ have the same char. and min. pol's, but they obviously (why?) aren't symetric.

But this is obvious if you already know about the Jordan Canonical Form, so if you don't then this won't help you...I guess.

Tonio

2. Diagonalizable matrices with the same characteristic polynomial are similar.

3. Diagonalizable matrices with the same minimal polynomial are similar.

4. Matrices with the same characteristic polynomial that factorize to different linear factors, are(matrices) similar.

5.Matrices with the same characteristic polynomial that factorize to different irreducible polynomial, are with the same minimal polynomial.