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I've got the following matrix $A$ for an endorphism within a base $v_1, v_2, v_3$

$$ A = \left( \begin{array}{ccc} 0 & 0 & -1 \\ 1 & 0 & -3 \\ 0 & 1 & -3 \\ \end{array} \right) $$

I need to find the base vectors $v^'_1, v^'_2, v^'_3$ for which the matrix of this endorphism looks like follows (this is the Jordan matrix of this endorphism as I worked out)

$$ A^' = \left( \begin{array}{ccc} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & -1 \\ \end{array} \right) $$

So I know there must exist a matrix $T$ so that $AT = TA^'$. But how does this help to get the base vectors $v^'_1, v^'_2, v^'_3$ based on $v_1, v_2, v_3$?

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    So far I found only solutions on the internet which use generalized eigenvectors and nilpotent etc. which are all terms we haven't learnt yet.. So I somehow need another way to find the base vectors.. So just if I knew matrix $A$ has another looking $B$ in another base independent of Jordan form etc.2012-05-01

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