<s>[INST] Eigenvalue and Eigenvector of $\small\pmatrix{0 & 0 \\ 0 & -7}$ I need help working out the eigenvectors for this matrix. 
$ \begin {pmatrix} 0 &  0 \\   0 & -7 \end{pmatrix} $ 
The original matrix is $ \begin {pmatrix} 5 &  0 \\   0 & -2 \end{pmatrix} $ , eigenvalues are 5,-2,
but I am not sure how to about the eigenvectors, 
as for 5 
$ \begin {pmatrix} 0 &  0 \\   0 & -7 \end{pmatrix} $ $ \begin{pmatrix} x \\ y \end{pmatrix}$ = $ \begin{pmatrix} 0 \\ 0 \end{pmatrix}$
from the first equation, $x$ and $y$ are both zero, but from the second equation $y = 0$, so what is the eigenvector? 
 [/INST] No, from the first equation, $x$ and $y$ are free. From the second equation, $y=0$. So your eigenvector is 
$$
\begin{bmatrix}1\\0 \end{bmatrix}
$$
as you can check, the equation is satisfied.
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