# Math Help - Eigenspace

1. ## Eigenspace

I have a question asking me to "find the eigenvalues and provide the basis of each eigenspace."

I don't have what an "eigenspace" is in my notes (we haven't talked about it in our lectures) and it's not in my textbook, either.

Could someone please explain what they're looking for?

2. Originally Posted by Thomas
I have a question asking me to "find the eigenvalues and provide the basis of each eigenspace."

I don't have what an "eigenspace" is in my notes (we haven't talked about it in our lectures) and it's not in my textbook, either.

Could someone please explain what they're looking for?
the eigenspace is the solution space of $( \lambda I - A) \bold{x} = \bold{0}$

i believe you can think of the basis for the eigenspace as the basis for the space spanned by the eigen vectors, but don't quote me on that. just go with what i said above

3. What do you mean by "solution space"?

4. Originally Posted by Thomas
What do you mean by "solution space"?
the solution vector for each eigen value. just find the eigen vectors as usual and then you can get the basis from them

for instance, if you find that the eigen vectors are of the form ${-t \choose t}$ then ${-1 \choose 1}$ is the basis, it is the solution vector without the parameter (you factor out the parameters)

5. Okay, so my eigenvalue is 1. It has multiplicity 3.

$det(xI-A)=\left( \begin{array}{ccc} x+3 & 0 & 8 \\ -4 & x-1 & -8 \\ -2 & 0 & x-5 \end{array} \right)$

After plugging in my eigenvalue, I find all three rows are equal, so I cancel out two of them. I'm left with:

$-2x+0y-4z=0$

I find the eigenvector to be:

$t\left( \begin{array}{ccc} -2 \\ 0 \\ 1 \end{array} \right)$

The question asks for the eigenvalue (which is 1) and the eigenspace. I still don't understand what the eigenspace is? It also has room for two different answers (of 3 rows and 1 column.)

I tried entering $\left( \begin{array}{ccc} -2 \\ 0 \\ 1 \end{array} \right)$ but it rejected it.

Help!

6. Originally Posted by Thomas
Okay, so my eigenvalue is 1. It has multiplicity 3.

$det(xI-A)=\left( \begin{array}{ccc} x+3 & 0 & 8 \\ -4 & x-1 & -8 \\ -2 & 0 & x-5 \end{array} \right)$

After plugging in my eigenvalue, I find all three rows are equal, so I cancel out two of them. I'm left with:

$-2x+0y-4z=0$

I find the eigenvector to be:

$t\left( \begin{array}{ccc} -2 \\ 0 \\ 1 \end{array} \right)$

The question asks for the eigenvalue (which is 1) and the eigenspace. I still don't understand what the eigenspace is? It also has room for two different answers (of 3 rows and 1 column.)

I tried entering $\left( \begin{array}{ccc} -2 \\ 0 \\ 1 \end{array} \right)$ but it rejected it.

Help!
Maybe you should post the original question.

-Dan