I'm trying to find the Jordan Normal Form of the following matrix:
$\pmatrix{2 & 0 & 1 & 1 \\0 & 2 & 1 & 1\\0 & 0 & 2 & 1\\0 & 0 & 0 & 2\\}$.
Now since its upper triangular I know that the only eigenvalue is 2 with algebraic multiplicity equal to $4$. Then by calculating $\text{rank}(A-2I)=2$, I can find the geometric multiplicity which is $\text{gm}=4-2=2$. Hence I will have two Jordan Block with dimensions summing to $4$.
Which of the following should I choose:
$J_3(2) \oplus J_1(2)$ or $J_2(2) \oplus J_2(2)$?
Two possible cases for JNF of a matrix
jordan-normal-formlinear algebramatrix-rank
Best Answer
One way is to find the minimal polynomial of the matrix. The multiplicity of $2$ as a root of that polynomial is the size of the largest block of that eigenvalue in the Jordan Normal Form. That will give you the size of the second block as well.