- #1
anastasis
- 3
- 0
Homework Statement
How do I show that the cp is p(x)=x^n, dimA=n?
Homework Equations
A^k=0 for some k (obviously need to show k=n); p(A)=0
The Attempt at a Solution
p(A)=0 <=> A^n + ... + det(A)=0
anastasis said:i see that p(A)=0 and that p(A)=A^n +...+det(A) BUT I'm still a bit confused by some details. A is nilpotent <=> A^k=0 ... must show k=n for one thing.
A characteristic polynomial for a nilpotent matrix is a polynomial equation whose roots are the eigenvalues of the matrix. It is used to determine the eigenvalues and the diagonalizability of the matrix.
The characteristic polynomial for a nilpotent matrix is calculated by taking the determinant of the matrix (A-λI), where A is the matrix and λ is an indeterminate. This will result in a polynomial with degree n, where n is the size of the matrix.
The degree of the characteristic polynomial is equal to the size of the matrix. This means that for a 3x3 matrix, the characteristic polynomial will be of degree 3, and for a 4x4 matrix, the characteristic polynomial will be of degree 4.
No, a nilpotent matrix can only have one characteristic polynomial. This is because the characteristic polynomial is determined by the matrix itself and its size, and a matrix cannot have two different sizes.
The characteristic polynomial for a nilpotent matrix can be used in various applications, such as in linear algebra, differential equations, and physics. It is used to determine the eigenvalues and diagonalizability of a matrix, which can provide important information about the behavior and properties of a system.