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If $A$ is a $3\times 3$ complex matrix with characteristic polynomial $f_A(t)=t^3-2$ then
$A$ is not diagonalizable.
$A$ has one eigenvalue with corresponding eigenspace that is $1$-dimensional and another eigenvalue with corresponding eigenspace that is $2$-dimensional.
$A$ has thee distinct eigenspaces, that are each $1$-dimensional.
it is impossible to determine if $A$ is diagonalizable without more information.