Solution EE.C26 Chris Black

Since we are given that the characteristic polynomial of $A$ is $\charpoly{A}{x} = (4-x)(1-x)^2$, we see that the eigenvalues are $\lambda = 4$ with algebraic multiplicity $\algmult{A}{4} = 1$ and $\lambda = 1$ with algebraic multiplicity $\algmult{A}{1} = 2$. The corresponding eigenspaces are \begin{align*} \eigenspace{A}{4} &= \spn{\colvector{1\\1\\1}} & \eigenspace{A}{1} &= \spn{\colvector{1\\-1\\0},\colvector{1\\0\\-1}} \end{align*}