Determine the characteristic polynomial

[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!

 

[Dick]

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.

 

[micromass]

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!