Here is the example I encountered : A matrix M(5\times 5) is given and its minimal poly

Here is the example I encountered :
A matrix ${\displaystyle M(5\times 5)}$ is given and its minimal polynomial is determined to be ${\displaystyle {(x-2)}^3}$. So considering the two possible sets of elementary divisors
${\displaystyle \{{(x-2)}^3,{(x-2)}^2\}\text{and}\{{(x-2)}^3,(x-2),(x-2)\}}$
we get two possible Jordan Canonical forms of the matrix , namely ${\displaystyle J_1}$ and ${\displaystyle J_2}$ respectively. So ${\displaystyle J_1}$ has 2 and ${\displaystyle J_2}$ has 3 Jordan Blocks.
Now we are to determine the exact one from these two. From the original matrix M, we determined the Eigen vectors and 2 eigen vectors were linearly independent. So the result is that ${\displaystyle J_1}$ is the one .So, to determine the exact one out of all possibilities , we needed two information -
1) the minimal polynomial ,together with 2) the number of linearly independent eigen vectors.
Now this was a question-answer book so not much theoretical explanations are given . From the given result , I assume the number of linearly independent eigen vectors -which is 2 in this case - decided ${\displaystyle J_1}$ to be the exact one because it has 2 Jordan Blocks. So the equation
"Number of linearly independent eigen vectors=Number of Jordan Blocks"
must be true for this selection to be correct .
Now this equation is not proved in this book or the text book I have read says nothing of this sort
So, that is my question here : How to prove the equation "Number of linearly independent eigen vectors=Number of Jordan Blocks"?