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I have a linear transformation on a 4 dimensional vector space with eigenvalue 2 and multiplicity 4. I want to get all the possible Jordan Forms of this linear transformation. Can anyone help me with this? Thanks.

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2 Answers 2

Taking a block of size 1 to mean no off-diagonal term, your are asking for all partitions of 4 into blocks: $$ 4 \models 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 $$ The first 4 means 3 1's, the final 1+1+1+1 means no off-diagonal entries.

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You just need to take combinations of Jordan blocks:

$$ \pmatrix{2},\;\pmatrix{2&1\\0&2},\;\pmatrix{2&1&0\\0&2&1\\0&0&2},\;\pmatrix{2&1&0&0\\0&2&1&0\\0&0&2&1\\0&0&0&2}. $$

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