Which vector generates eigenspace corresponding to particular eigenvalue

Question

So I have calculated eigenvalues for matrix A which are 2,1,0,-1,-3, but I'm finding hard to solve how to find vector which generates the eigenspace corresponding to eigenvalue 2

eigenvalue and eigenspace

Answer 1

As you have five different eigenvalues for a $5 \times 5$ matrix, all eigenspaces will be generated by a single vector.

It then suffices to check which of the listed vectors $\vec{v}$ is the eigenvector with eigenvalue $2$. You do this by calculating $A \cdot \vec{v}$ and checking if $A \cdot \vec{v} = 2 \vec{v}$.

If the answers weren't provided, you would have to solve the linear system

$$ A \cdot \vec{v} = 2 \vec{v}.$$

Answer 2

Computing all of the eigenvalues of the given matrix builds character, but it’s way more work than necessary in order to solve this problem. You’re basically being asked to determine which of the given vectors satisfies the eigenvector equation $Av=2v$, so all you need to do is check the five products. You can further save yourself some work by working with only one row of $A$ to do some culling. The third row has a zero in it, so using that one should save a bit of work. Multiplying that row by answer A produces $-12$, so that can’t be the correct one. Doing this with the other three possible answers should leave you with only one candidate, and you should in principle verify that it’s really the correct answer by computing $Av$ in its entirety for that vector.

Alternatively, you could take the eigenvalue $2$ as a given, compute the null space of $A-2I$ using your favorite method, and compare that to the potential solutions, but I think that will be more work than simply checking those four answers directly.