Adding Eigenvectors

Question

I am trying to understand the answer to the following problem.

If $v_1 = \begin{bmatrix} 3\\ 3\end{bmatrix} $ and $ v_2 = \begin{bmatrix}-2\\ -3\end{bmatrix} $ are eigenvectors of matrix $A$ corresponding to the eigenvalues $\lambda_1 = -5$ and $\lambda_2 = -4$, respectively,

then $A(v_1 + v_2) = \begin{bmatrix}\_\\ \_\end{bmatrix}$ and $A(-3v_1) = \begin{bmatrix}\_\\ \_\end{bmatrix}$

Through some guidance, I have found the answers to be $A(v_1 + v_2) = \begin{bmatrix}-7\\ -3\end{bmatrix}$ and $A(-3v_1) = \begin{bmatrix}45\\ 45\end{bmatrix}$

I was told that to get the first answer, I needed to multiply $v_1$ by $\lambda_1$ and $v_2$ by $\lambda_2$, then add these two vectors.

To find the second answer I was told to multiply $v_1$ by $\lambda_1$ and then multiply by $-3$.

So I am wondering:

• What the $A$ in $A(v_1 + v_2)$ and $A(-3v1)$ means
• Why I need to multiply the eigenvectors by the eigenvalues

Thanks.

Answer 1

The definition of eigenvector and eigenvalue: $Av=\lambda v$.

If you know which are the eigenvalues and the eigenvectors, you can operate so: $$A(v1+v2)=A(v1)+A(v2)=\lambda v1 + \lambda v2$$ $$A(-3v1)=-3A(v1)=-3\lambda v1$$

Sustitute and calculate!