Characteristic Polynomials and Nilpotent Operators

[glacier302]

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 : )

 

[micromass]

You could try the theorem of Cayley-Hamilton. This will answer your question...

 

[glacier302]

Thank you, that makes it very easy...if only I could remember when to use these theorems on my own!

 

[glacier302]

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?

 

[micromass]

Well, firstly, how would you define the characteristic polynomial in an infinite-dimensional space??

 

[glacier302]

That's a good point!