1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Can you look up the defining properties of a linear transformation? $\endgroup$ – Alex Pavellas Dec 9 '16 at 23:36
  • $\begingroup$ Read the first sentence of your problem to find what $A$ means. As for the second half of your question, you should make sure you know what eigenvalues and eigenvectors are. $\endgroup$ – Cameron Buie Dec 9 '16 at 23:52
4
$\begingroup$

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!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.