Are linear combinations of eigenvectors also a eigenvector of a matrix?

Question

If $A$ is a square matrix, $v$ and $w$ are eigenvectors of $A$, then $v + w$ is also an eigenvector of $A$.

Would this statement hold true? If not, could you give a counter example?

Answer 1

If $v$ and $w$ are eigenvectors for the same eigenvalue, then the sum $v+w$ is also an eigenvector. Otherwise it is false

For example, consider $A=\text{diag}\{1,2\}$, a $2 \times 2$ matrix. Then $(1,0)^t$ and $(0,1)^t$ are eigenvectors for the eigenvalues $1$ and $2$ respectively,but $(1,1)^t$ is not an eigenvector

Answer 2

No. Eigenvectors of matrix \begin{pmatrix}2&1\\1&2\end{pmatrix}

are, for example $$\color{blue} {\begin{pmatrix} 1\\1\end{pmatrix}} \text{, }\color{violet} {\begin{pmatrix} -1\\1\end{pmatrix}}$$

(because the transformation determined by this matrix don't change their direction)

but their sum

$$\color{blue} {\begin{pmatrix} 1\\1\end{pmatrix}} + \color{violet} {\begin{pmatrix} -1\\1\end{pmatrix}} = {\begin{pmatrix} 0\\2\end{pmatrix}}$$

not.