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I have the two matrices:

$\begin{pmatrix}1&-4&-2\\ 0&1&0\\ 0&4&3\end{pmatrix}$ and $\begin{pmatrix}3&0&0\\ \:0&1&1\\ \:0&0&1\end{pmatrix}$

I know they have the same trace and determinant but I know that isn't enough to prove they are similar... what are the next steps I should take? Is proving they have the same eigenvalues enough to show they are similar?

user520403
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    The first matrix is already in Jordan normal form... – user251257 Feb 28 '18 at 20:02
  • Do you have studied matrix reduction? –  Feb 28 '18 at 20:03
  • For characteristic $2$ the first matrix is diagonal, but the second is not diagonalizable. Hence they cannot be similar in this case. – Dietrich Burde Feb 28 '18 at 20:10
  • And spoiler: they aren't similar. Just compute the Jordan normal form. – user251257 Feb 28 '18 at 20:10
  • @DietrichBurde nope. The first matrix - $I$ has kernel dimension 2, unlike the second one. – user251257 Feb 28 '18 at 20:15
  • @DietrichBurde $(0,-4,-2)$ and $(0,4,2)$ look pretty linearly dependent to me. – user251257 Feb 28 '18 at 20:20
  • We haven't learned Jordan normal form yet. All we learned is B=P^−1AP and I'm not sure how to come up with a P (or I guess since they aren't similar there won't be a P? I'm not sure how to prove that) – user520403 Feb 28 '18 at 20:25
  • @user520403 You do not need Jordan form. Just compute the system of linear equations with the $9$ variables being the entries of $P$ coming from $PA=BP$. Then you get $\det(P)=0$ from it, so they are not similar. – Dietrich Burde Feb 28 '18 at 20:31

3 Answers3

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The first matrix is diagonalizable, namely $P^{-1}AP={\rm diag}(1,1,3)$ with $$ P=\begin{pmatrix} -3 & 2 & -2\cr -2 & 1 & 0 \cr 4 & -2 & 2 \end{pmatrix}. $$ The second matrix is not diagonalizable, so they are in fact not similar.

Dietrich Burde
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Hint: The definition of similarity between matrices is the following:

Two square matrices of the same dimensions A and B are said to be similar if there is a matrix P such that $$B = P^{-1}AP$$

Try finding a matrix $P$ for your exercise.

Skyris
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Hint: A matrix $A$ and $B$ are said to be similar if $$B=P^{-1}AP$$ for some invertible matrix $P$. This link: How do I tell if matrices are similar? outlines the entire process.

Dragonite
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