Find a non-singular matrix $P$ such that $A=PBP^{-1}$ if $A=\left[ \begin {array}{ccc} 2&1&3\\ 1&5&4\\ 3&4&6\end {array} \right]$ and $B=\left[ \begin {array}{ccc} 5&4&5\\3&3&4 \\1&3&5\end{array} \right]$.
What I did :
$A$ and $B$ are similar. Indeed $\chi_A=\chi_B={X}^{3}-13\,{X}^{2}+26\,X-1$. Indeed the polynomial $\chi_A=\chi_B$ has three real roots. So the matrices $A$ and $B$ are both similar to the same digonal matrix $D$, so $A$ and $B$ are similar. Then there exists $P$ such that $A=PBP^{-1}$. When I try to compute the eigenvalues and eigenvectors, it's quite complicated, so I wonder if there is another way...