Can two different matrices have the same eigenvectors? Assume that we have two symmetric and non-negative matrices $A$ and $B$. They are not the same, even if they might look the same to the untrained eye. If I want to find the eigenvectors, I could use the $V$ matrix from the SVD
$$A = USV^T$$
where matrix $V$ holds the eigenvectors in its columns.
Are there any situations where the eigenvectors of $V$ from the $A$ matrix, can be the same as $V$ matrix from the $B$ matrix?

Take as $A=\begin{bmatrix}
1 & 0 \\
 0& 1 \\
\end{bmatrix}$ and $B= \begin{bmatrix}
0 & 0 \\
 0& 1 \\\end{bmatrix}$. $A$ has eigenvalue $1$ and eigenvector
$v=\begin{bmatrix}
v_{1} \\v_{2}
\end{bmatrix}=\begin{bmatrix}
0 \\1
\end{bmatrix}$ which is also an eigenvector of $B$ and also the other eigenvector is $\begin{bmatrix}
1 \\0
\end{bmatrix}$.
Now $B$ has also eigenvalue $0$ and $B\begin{bmatrix}
1 \\0
\end{bmatrix}$=$\begin{bmatrix}
0 \\0
\end{bmatrix}$.
So $A,B$ are two non-negative symmetric different matrices which have the same eigenvectors! Therefore the answer is yes they can!!
