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Write down all possible Jordan normal forms for matrices with characteristic polynomial$ (x − λ)^5$.
In each case, calculate the minimal polynomial and the geometric multiplicity of the eigenvalue λ.
For the only eigenvalue $\lambda$, the possible JNF is just assigned 1 to every column above the diagonal since the min polynomial can be any degree from 1 to 5?
I figured out the possible JNF using the possible minimal polynomial
$(x-\lambda)$and $(x-\lambda)^2$ and $(x-\lambda)^3$...$(x-\lambda)^5$ In total, it is 7 possibility (corresponding to each minimal polynomial and $\lambda$ has to appear 5 times.) But I don't understand that why the number of blocks gives the geometric multiplicity since each represent one eigenspace.
Thank you so much!