How does the size of Jordan blocks in the Jordan canonical form of matrix A relate to exponents $d_i$ in the factorization of the minimal polynomial $m_A(x) = (x-\lambda_1)^{d_1}(x-\lambda_2)^{d_2}\ldots(x-\lambda_m)^{d_m}$?
What does it say about the diagonalizability of A?
The reason I'm posting is that I'm out of my depth completely. What I can surmise is there will be at least one Jordan block the size of a particular exponent $d_i$. Please, if anyone decides to answer, provide at least a short explanation as I really want to understand this better. Thank you very much.
Best Answer
$d_i$ is the size of the largest Jordan block associated with $\lambda_i$. $A$ is diagonalizable if and only if all $d_i$ are equal to $1$ (i.e. there are no Jordan blocks with size larger than $1$).