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I am trying to solve this problem. If $A$ is a $2 \times 2$ matrix with complex entries, then $A$ is similar over $\Bbb C$ to a matrix of one of the two types
$$ M= \left[ {\begin{array}{cc} a & 0\\ 0 & b\\ \end{array} } \right], $$ $$ M= \left[ {\begin{array}{cc} a & 0 \\ 1 & a \\ \end{array} } \right]. $$

Could you please tell me how to start? I have no idea! Thanks

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    $\begingroup$ Hint: use Jordan normal form. $\endgroup$ – TZakrevskiy Mar 3 '14 at 14:47
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Break into cases by the number of possible eigenvalues.

If $M$ is diagonal or has two distinct eigenvalues it is diagonalizable, so is similar to the first type. Otherwise there is only a single 1-dimensional eigenspace and it is similar to the second type.

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