# How can I compute $A(v_1 + v_2)$ where $v_1$ and $v_2$ are eigenvectors of the matrix A

If $$v_1 = \begin{bmatrix}5\\3\end{bmatrix}$$ and $$v_2 = \begin{bmatrix}3\\1\end{bmatrix}$$ are eigenvectors of a matrix $$A$$ corresponding to the eigenvalues $$\lambda_1 = -1$$ and $$\lambda_2 = 4$$ respectively, then

$$A(v_1 + v_2)= ?$$

What about $$A(3v_1) = ?$$

I get what I'm doing wrong. I did $$v_1 + v_2 = \begin{bmatrix}5\\3\end{bmatrix} + \begin{bmatrix}3\\1\end{bmatrix} = \begin{bmatrix}8\\4\end{bmatrix}$$ then I did $$A(\begin{bmatrix}8\\4\end{bmatrix})= \begin{bmatrix}-1&0\\0&4\end{bmatrix}*\begin{bmatrix}8\\4\end{bmatrix} = \begin{bmatrix}-8\\16\end{bmatrix}$$

• Hint. Do not add up the vectors first. You must first examine the multiplication $Av=\lambda{v}$ on both $v1$ and $v2$. And your eigenvalues are given.... Jul 19, 2016 at 2:30

$A$ is only diagonal with respect to a basis of eigenvectors, namely $v_1$ and $v_2$, not the standard basis, which is why your computation fails. Try using linearity and the definition of eigenvectors instead:
$A(v_1+v_2)=Av_1+Av_2=\lambda_1 v_1+\lambda_2 v_2$
Similarly for $A(3v_1)$.
• You must have made an error. Did you compute $-v_1+4v_2$? Jul 19, 2016 at 2:39