Characteristic Polynomials and Nilpotent Operators

In summary, the conversation discusses whether an operator T is nilpotent if its characteristic polynomial is (-1)^n*t^n and how to prove this fact. The theorem of Cayley-Hamilton is suggested as a possible solution, but there is a question about its applicability to infinite-dimensional vector spaces. The conversation ends with the realization that the definition of characteristic polynomial in an infinite-dimensional space may need to be clarified before applying the theorem.
  • #1
glacier302
35
0
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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  • #2
You could try the theorem of Cayley-Hamilton. This will answer your question...
 
  • #3
Thank you, that makes it very easy...if only I could remember when to use these theorems on my own!
 
  • #4
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?
 
  • #5
Well, firstly, how would you define the characteristic polynomial in an infinite-dimensional space??
 
  • #6
That's a good point!
 

1. What is a characteristic polynomial?

A characteristic polynomial is a polynomial that is associated with a square matrix. It is found by taking the determinant of the matrix minus a variable, and solving for the variable. The resulting polynomial is called the characteristic polynomial because its roots, or solutions, are the eigenvalues of the matrix.

2. How are characteristic polynomials and eigenvalues related?

The roots, or solutions, of a characteristic polynomial are the eigenvalues of the matrix. This means that by finding the characteristic polynomial and solving for the variable, we can determine the eigenvalues of the matrix.

3. What is a nilpotent operator?

A nilpotent operator is a linear operator on a vector space that satisfies the property that some power of the operator equals the zero operator. In other words, if we repeatedly apply the operator to a vector, it will eventually become zero. This is why it is called a "nilpotent" operator.

4. What is the relationship between nilpotent operators and characteristic polynomials?

For a nilpotent operator, the characteristic polynomial will be of the form (x^n) where n is the dimension of the vector space. This is because applying the operator n times will result in the zero operator, and the determinant of the zero matrix is 0. Therefore, the characteristic polynomial will have n roots, all of which are 0. This also means that the only eigenvalue of a nilpotent operator is 0.

5. How can characteristic polynomials and nilpotent operators be used in real-world applications?

Characteristic polynomials and nilpotent operators are used in a variety of fields, including physics, engineering, and computer science. They can be used to solve systems of differential equations, analyze the stability of systems, and find the Jordan canonical form of a matrix. In computer science, they are used in algorithms such as the QR algorithm for finding eigenvalues and eigenvectors of large matrices. They are also important in understanding the behavior of matrices and operators in various applications.

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