Find Eigenvalues & Eigenspace for (3,0) (8,-1) Matrix | Homework Statement

[Chewybakas]

Homework Statement

Find the eigenvalues and eigenspace of the given vector.

Homework Equations

Matrix = (3,0)
(8,-1)

The Attempt at a Solution

I have determined the eigenvalues to be -1 and 3, but when I try compute the eigenspace when lambda = -1 I constantly get confused and end up with the space equal to span(t[0,0]) tεℝ. Can anyone help or confirm that answer!

 

[micromass]

I agree with the eigenvalues. Can you show your computations for the eigenspace for $\lambda = -1$?

 

[Mark44]

Homework Statement

Find the eigenvalues and eigenspace of the given vector.

Homework Equations

Matrix = (3,0)
(8,-1)

The Attempt at a Solution

I have determined the eigenvalues to be -1 and 3, but when I try compute the eigenspace when lambda = -1 I constantly get confused and end up with the space equal to span(t[0,0]) tεℝ. Can anyone help or confirm that answer!

What are the equations you get when you try to find the eigenvectors for λ = -1?

 

[Chewybakas]

I reduced the original matrix to reduced row echelon form which gave me the identity 2x2 matrix, which when finding two values v1,v2 where when multiplied by the identity matrix gives zero, I get the answer stated above but when i tried a different way I get the eigenspace 2v1 and 9v1 which again confuses me.

 

[Mark44]

I reduced the original matrix to reduced row echelon form which gave me the identity 2x2 matrix, which when finding two values v1,v2 where when multiplied by the identity matrix gives zero, I get the answer stated above but when i tried a different way I get the eigenspace 2v1 and 9v1 which again confuses me.

That's not how it works.

For λ = -1, you are solving the equation (A - λI)x = 0
For this eigenvalue, A - λI is
$$ \begin{bmatrix} 4 & 0 \\ 8 & 0\end{bmatrix}$$

When you row reduce this you DON'T get the identity matrix.