Here is the example I encountered :A matrix M(5×5) is given and its minimal polynomial is determined to be (x−2)3. So considering the two possible sets of elementary divisors{(x−2)3,(x−2)2} and {(x−2)3,(x−2),(x−2)}we get two possible Jordan Canonical forms of the matrix , namely J1 and J2 respectively. So J1 has 2 and J2 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 J1 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 J1 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"?

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Here is the example I encountered :A matrix  M(5×5) is given and its minimal polynomial is determined to be (x−2)3. So considering the two possible sets of elementary divisors{(x−2)3,(x−2)2}  and  {(x−2)3,(x−2),(x−2)}we get two possible Jordan Canonical forms of the matrix , namely J1 and J2 respectively. So J1 has 2 and J2 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 J1 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 J1 to be the exact one because it has 2 Jordan Blocks. So the equation&quot;Number of linearly independent eigen vectors=Number of Jordan Blocks&quot;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 &quot;Number of linearly independent eigen vectors=Number of Jordan Blocks&quot;?

Brody Buckley · Accepted Answer

Let A be a matrix reduced in Jordan form. Notice that for every Jordan block relative to the eigenvalue α there is an eigenvector with respect to α and these vectors ar independent. To see this just look at the first column of a generic block. So Jordan blocks independent eigenvectors. To prove the other inequality, consider the matrix A−αI for every eigenvalue α and calculate its rank.

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

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