<s>[INST] How to find a matrix $C$ such that $C^{-1}AC$ is in Jordan block form. $A:=\begin{bmatrix}
6 & -1\\ 
4 & 2
\end{bmatrix}$
Now, just to show I've done some working, at least to find $A$'s eigenvalues and deduced that it's not diagonalisable:

Any hints/advice?
Thanks
 [/INST] For a $2 \times 2$ (say complex) matrix $A$ which is not diagonalizable, there is a method to determine the conjugating matrix $C$ such that $C^{-1}AC$ is in Jordan form. The only way that $A$ can fail to be diagonalizable is if the characteristic polynomial of $A$ has a repeated root $s,$ but $A \neq sI.$ Then the only possibility for the Jordan form is $\left( \begin{array}{clcr} s & 1\\0&s \end{array} \right)$. In order to have trace $2s$ and determinant $s^{2},$ the matrix $A$ must either  have the form $\left( \begin{array}{clcr} s+a &  ta\\\frac{-a}{t} & s-a \end{array} \right)$ for some $a,t \neq 0,$ or have one of the forms $\left( \begin{array}{clcr} s & t\\0&s \end{array} \right)$ or $\left( \begin{array}{clcr} s & 0\\t&s \end{array} \right)$ for some non-zero $t.$ We have to solve the equation $AC = C \left( \begin{array}{clcr} s & 1\\0&s \end{array} \right)$ with ${\rm det} C \neq 0.$ This reduces to solving one of the equations $\left( \begin{array}{clcr} a &  ta\\\frac{-a}{t} & -a \end{array} \right)C = C\left( \begin{array}{clcr} 0 & 1\\0&0 \end{array} \right)$ or $\left( \begin{array}{clcr} 0 &  t\\0 & 0 \end{array} \right)C = C\left( \begin{array}{clcr} 0 & 1\\0&0 \end{array} \right)$ or $\left( \begin{array}{clcr} 0 &  0\\t & 0 \end{array} \right)C = C\left( \begin{array}{clcr} 0 & 1\\0&0 \end{array} \right).$ This can always be done.
In your case, you have $s = 4, a = 2$ and $t = \frac{-1}{2}.$
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