# Non-similarity of possible Jordan forms

Full exercise: Determine all possible Jordan canonical forms (up to the ordering of the Jordan blocks) for a 6x6 matrix A, if A has eigenvalue 2 with algebraic multiplicity 6, and geometric multiplicity 3. Explain why the obtained Jordan canonical forms are not similar.

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Let $J_k(\lambda)$ denote the Jordan block of size $k$ associated with $\lambda$. Let $\oplus$ denote the direct sum. The possible Jordan forms (up to permutation) are $$A_1 = J_4(2) \oplus J_1(2) \oplus J_1(2)\\ A_2 = J_3(2) \oplus J_2(2) \oplus J_1(2)\\ A_3 = J_2(2) \oplus J_2(2) \oplus J_2(2)\\$$ To show that these matrices are similar, it suffices to find a property of one that applies to all similar matrices but fails to apply to the others.
For example: $(A_3 - 2I)^2 = 0$, and any matrix $A$ similar to $A_3$ would also satisfy $(A_3 - 2I)^2 = 0$. However, $(A_2 - 2I)^2 \neq 0$.
Similarly, $(A_2 - 2I)^3 = 0 \neq (A_1 - 2I)^3$.