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What does the following mean --

"The Jordan Canonical Form of the operator $w{d\over dw}$ acting on the complex vector space of polynomials in $w$ of degree less than $n$"?

Thank you.

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    Which part do you not understand?2012-05-19
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    @ChrisEagle: I understand what a complex vector space of polynomials in $w$ of degree less than $n$ means, but not the bit before it. I know how a JCF matrix looks like, but how do you turn an operator into a matrix??2012-05-19
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    $w\frac{d}{dw}$ is a linear map on a comple vector space, the composition of multiplication by $w$ and $\frac{d}{dw}$ so it has a Jordan form, it is a sum of a semisimple linear map and a nilpotent, and in an appropriate basis this is called a Jordan normal form. I guess you should rather ask what that form is. It is probably something very close to the identity, when you differentiate you lower by one power of $w$ and multiply by an integer, so you revert part of that multiplying by $w$.2012-05-19
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    I should have said "Jordan decomposition" instead of "Jordan form", or "Jordan-Chevalley decomposition". This refers to the sum decomposition. The "normal form" is actually writing the corresponding triangular matrix in a basis of generalized eigenvectors.2012-05-19

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