Hello,

the task is following:

Let's assume that vector $\displaystyle \bar v$ is eigenvector corresponding to eigenvalue $\displaystyle \lambda_{A}$ of matrix $\displaystyle A$. Also $\displaystyle \bar v$ corresponds to eigenvalue $\displaystyle \lambda_{B}$ of matrix $\displaystyle B$. Now the questions:

I Is the $\displaystyle \bar v$ eigenvector of matrix $\displaystyle A+B$?

II Is the $\displaystyle \bar v$ eigenvector of matrix $\displaystyle AB$?

If answer is yes to atleast one question, it must be shown, to which eigenvalue vector $\displaystyle \bar v$ corresponds.

I think I should use following theorem:

"If $\displaystyle \bar v$ is eigenvector corresponding to eigenvalue of matrix $\displaystyle A$, then $\displaystyle \bar v$ is is eigenvector corresponding to $\displaystyle A^k$'s $\displaystyle \lambda^k$".

But somehow, I can't get anywhere, though I inspected eigenvalues and -vectors definitions closely. Also it's no problem to me calculate those vectors and values in normal cases. So, any help is appreciated. Thank you!