# Determine the characteristic polynomial

• LeakyFrog
In summary, finding a basis for eigenspaces using only the eigenvalues is impossible as the eigenspaces and their bases are not determined solely by the eigenvalues. It is necessary to know the matrix (or at least more information) to determine the eigenvectors.
LeakyFrog
Hey I'm studying for an exam and one of the things i need to know is this:

4. Given the eigenvalues of a matrix:
a) Determine the characteristic polynomial.
b) Find vectors than can act as bases for the associated eigenspaces.

Part a seems relatively straight forward but for part b I wondering if you need to be given a matrix along with the eigenvalue. Or is there a way where if you are just given eigenvalues that you can find bases for eigenspaces without the matrix. Almost seems impossible but i just figured I would ask.

Thanks!

Yes, it's impossible. Any vector can have any eigenvalue if you choose the right matrix. You really do have to use the matrix to find the eigenvalues and eigenvectors.

Hi LeakyFrog!

You are absolutely correct. To find a basis for the eigenspaces, you need to know the matrix. A simple example is this:

$$A=\left(\begin{array}{cc} 0 & 1\\ 0 & 0\\\end{array}\right)$$

Then the eigenvalues of A are 0 and 0, and the eigenspace is generated by the vector (1,0).

However, the zero matrix also has eigenvalues 0 and 0, but the eigenspace is generated by the vectors (1,0) and (0,1).

So, as you see, not even the dimension of the eigenspaces is determined by the eigenvalues, thus the eigenspaces and their bases are also not determined by the eigenvalues.

So in short: you need to know the matrix (or at least some more information) to know something about the eigenvectors!

## 1. What is the characteristic polynomial?

The characteristic polynomial is a polynomial equation that is used to find the eigenvalues of a square matrix. It is formed by taking the determinant of the matrix's difference from its identity matrix and setting it equal to zero.

## 2. How is the characteristic polynomial used in linear algebra?

The characteristic polynomial is used to find the eigenvalues of a square matrix, which are important in solving systems of linear equations and understanding the behavior of linear transformations. It can also be used to determine the diagonalizability of a matrix.

## 3. Can the characteristic polynomial be used to find the characteristic equation?

Yes, the characteristic polynomial is the same as the characteristic equation. Both terms refer to the polynomial equation used to find the eigenvalues of a square matrix.

## 4. How does the degree of the characteristic polynomial relate to the size of the matrix?

The degree of the characteristic polynomial is equal to the size of the matrix. For example, a 3x3 matrix will have a characteristic polynomial of degree 3. This is because the determinant of a matrix with n rows and columns will be a polynomial of degree n.

## 5. Can the characteristic polynomial be used for non-square matrices?

No, the characteristic polynomial is only applicable to square matrices. Non-square matrices do not have eigenvalues and therefore do not have a characteristic polynomial.

• Calculus and Beyond Homework Help
Replies
2
Views
570
• Calculus and Beyond Homework Help
Replies
19
Views
3K
• Calculus and Beyond Homework Help
Replies
4
Views
227
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• General Math
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
774
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
7
Views
3K
• Calculus and Beyond Homework Help
Replies
4
Views
1K