Finding eigenvectors for a matrix

Question

I have a problem finding eigenvectors for a matrix

The initial matrix is: $$A=\begin{bmatrix}-0.5&0.5\\0.14&-0.19\end{bmatrix}$$ Then i find the eigenvalues by using $det(A-\lambda * I)$

Which gives: $\lambda_1 = -0.652$ and $\lambda_2 = -0.038$

Then i try to do something like $(A-\lambda_{11}*I)V=0$ where $$V=\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$$

but I don't really get how to proceed.

I try to put up some kind of system like

$$0.152*v_1 + 0.5*v_2 = 0$$ $$0.14*v_1 + 0.462*v_2 = 0$$

but I can't seem to get anything useful out of it. If someone can show me how I can do the 1st I think I'll be able to do the 2nd one with $\lambda_2$

Answer 1

Well logically the second row equals the first one multiplied by a constant.

You need to choose $v_2$. For example, $v_2 =1$ and you deduce $v_1$ from it. It gives you the first eigenvector.

If you need to choose one coordinate, it's because the eigenvectors corresponding to a single eignevalue form a vector space.

Answer 2

Since $\lambda_1$ is eigenvalue the linear system will give you infinite number of eigenvectors, so what you need to do is to use the equations to find the span of the eigenspace dtermined by $\lambda_1$, i.e find the kernel of $A-\lambda_1I$, then just pick one vector in the kernel and you get an eigenvector corresponding to $\lambda_1$