I'm currently in the middle of a question where I'm given a 3×3 matrix:
$$\left(\begin{array}{rrr}
3 & 0 & 0\\
-2 & 7 & 0\\
4 & 8 & 1
\end{array}\right).$$
and have been asked to find the characteristic polynomial, eigenvalue and eigenvector/eigenspace when $\lambda = 3$
the aswer I got is $$ \left(\begin{array}{r}
\frac14\\
\frac18\\
1
\end{array}\right) $$
The question then asks to find the basis of this eigenspace, how do I go about that?
Best Answer
The vector you give is an eigenvector associated to the eigenvalue $\lambda = 3$. The eigenspace associated to the eigenvalue $\lambda = 3$ is the subvectorspace generated by this vector, so all scalar multiples of this vector.
A basis of this eigenspace is for example this very vector (yet any other non-zero multiple of it would work too).