$α :( a, b, c)^T → (a − b, a + 2b + c, −2a + b − c)^T$. Find the eigenvalues of $α$ and, for each eigenvalue, find the associated eigenspace.

Question

Define $α ∈ End(R^3)$ by $α :( a, b, c)^T → (a − b, a + 2b + c, −2a + b − c)^T$. Find the eigenvalues of $α$ and, for each eigenvalue, find the associated eigenspace.

I know that I need to solve $\alpha x=\lambda x$ where $\lambda$ is an eigenvalue and $x=(a,b,c)^T$. However, I can't cancel out any of the variables to solve explicitly for $\lambda$. Any solutions or hints are greatly appreciated.

Answer 1

Your eigenvectors are (-1,1,1),(-1,-2,7),(-1,0,1)

The related eigenspaces are sp(-1,1,1) sp(-1,-2,7) and sp(-1,0,1).

The eigenvalues are respectively 2, -1 and 1.