# If A is nxn nilpotent matrix, this char(A) = x^n

• Mr Davis 97
In summary, a nilpotent matrix is a square matrix where some power of the matrix is equal to the zero matrix. The characteristic polynomial of a nilpotent matrix is always of the form p(x) = x^n, where n is the size of the matrix. This polynomial can be useful in solving systems of linear equations by finding eigenvalues and eigenvectors, as well as providing information about the matrix's rank and determinate. A non-nilpotent matrix cannot have a characteristic polynomial of x^n, as it must have nonzero values.
Mr Davis 97

## Homework Statement

If ##A## is an ##n \times n## nilpotent matrix, then the characteristic polynomial of ##A## is ##x^n##

## The Attempt at a Solution

Suppose that ##A## has an eigenvalue with corresponding eigenvector such that ##A v = \lambda v##. Then ##A^k v = \lambda^k v = 0##, and since ##v \ne \vec{0}##, ##\lambda^k = 0 \implies \lambda = 0##. Since ##0## is the only eigenvalue, and since the characteristic polynomial must be of degree n, the characteristic polynomial must be ##x^n##.

You should introduce k. Apart from that: right.

Yes.

## 1. What does it mean for a matrix to be nilpotent?

A nilpotent matrix is a square matrix in which some power of the matrix is equal to the zero matrix. In other words, multiplying the matrix by itself a certain number of times will eventually result in a matrix of all zeros.

## 2. How is the characteristic polynomial of a nilpotent matrix related to its size?

The characteristic polynomial of a nilpotent matrix is always of the form p(x) = x^n. This means that the degree of the polynomial is equal to the size of the matrix (n x n).

## 3. Is every nilpotent matrix of size nxn guaranteed to have a characteristic polynomial of x^n?

Yes, for a nilpotent matrix of size nxn, the characteristic polynomial is always x^n. This is because the nilpotent property ensures that the matrix will eventually become the zero matrix when raised to the power of n.

## 4. Can a non-nilpotent matrix have a characteristic polynomial of x^n?

No, a non-nilpotent matrix cannot have a characteristic polynomial of x^n. If a matrix is not nilpotent, then there must be some nonzero values in the matrix, and therefore the polynomial cannot be of the form x^n.

## 5. How can the characteristic polynomial of a nilpotent matrix be useful in solving systems of linear equations?

The characteristic polynomial can be used to find the eigenvalues of a nilpotent matrix. These eigenvalues can then be used to find the eigenvectors, which can in turn be used to solve systems of linear equations. Additionally, the characteristic polynomial can provide information about the matrix, such as its rank and determinate, which can also be useful in solving systems of linear equations.

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