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Suppose we are given the  characteristic polynomial and minimal polynomial of a matrix $(x-a)^4(x-b)^2$ and $(x-a)^2(x-b)$ say. Then I can tell what the largest Jordan blocks are, and hence work out the possible forms the JNF can take. However, my question specifies that the matrix is "complex" whereas $a,b\in \mathbb R$. Is there some catch?

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    Matrices with real entries are a subset of matrices with complex entries - are you sure you're supposed to have non-real entries?2012-03-16
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    @MattPressland: Thant's what I thought, but I just want to make sure that there are no caveats :)2012-03-16
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    Since the characteristic polynomial factors into linear factors, there should be no problems - after all, all eigenvalues are real.2012-03-16
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    @Chump Fair enough! :) I don't think it's an issue, I regularly got given exercises to do with theorems about complex matrices in which everything could be done with integers, let alone general real numbers!2012-03-16
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    $(x-a)^2(x-b)$ cannot be a minimal polynomial. Minimal polynomials are products of distinct irreducible factors. These factors are always linear over $\mathbb{C}$ or any closed field, but can be linear or quadratic over $\mathbb{R}$. The minimal polynomial tells you which eigenvalues the matrix has (in the field). The characteristic polynomial tells you, additionally, their algebraic multiplicity, which is the sum of the sizes of all Jordan blocks corresponding to each eigenvalue. However, neither of these gives you the geometric multiplicity, which equals the number of such blocks.2012-03-16

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