Characteristic Polynomials and Nilpotent Operators

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Discussion Overview

The discussion revolves around the relationship between characteristic polynomials and nilpotent operators, specifically examining whether an operator T with a characteristic polynomial of the form (-1)^n*t^n can be classified as nilpotent. The scope includes theoretical considerations and the application of the Cayley-Hamilton theorem.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that if the characteristic polynomial of an operator T is (-1)^n*t^n, then T must be nilpotent, based on observations from specific matrix forms.
  • Another participant proposes using the Cayley-Hamilton theorem to address the question of nilpotency.
  • A later reply expresses uncertainty about the applicability of the Cayley-Hamilton theorem to infinite-dimensional vector spaces.
  • Another participant questions how the characteristic polynomial would be defined in the context of infinite-dimensional spaces.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the Cayley-Hamilton theorem, particularly in relation to infinite-dimensional spaces, indicating that the discussion remains unresolved regarding the broader implications of nilpotency in such contexts.

Contextual Notes

The discussion does not resolve the definitions or implications of characteristic polynomials in infinite-dimensional spaces, nor does it clarify the conditions under which the Cayley-Hamilton theorem applies.

glacier302
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If the characteristic polynomial of an operator T is (-1)^n*t^n, is T nilpotent?

My first instinct for this question is that the answer is yes, because the matrix form of T must have 0's on the diagonal and must be either upper triangular or lower triangular. This is what I found when I tried to find 2x2 and 3x3 matrices with characteristic polynomial (-1)^n*t^n. However, I'm not sure how to actually prove this fact (especially for the nxn case), or how to show that T is nilpotent using this fact.

Any help would be much appreciated : )
 
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You could try the theorem of Cayley-Hamilton. This will answer your question...
 
Thank you, that makes it very easy...if only I could remember when to use these theorems on my own!
 
One question: Doesn't the Cayley-Hamilton Theorem only apply to linear operators on finite-dimensional vector spaces? What if the vector space is infinite-dimensional?
 
Well, firstly, how would you define the characteristic polynomial in an infinite-dimensional space??
 
That's a good point!
 

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