Eigenvectors for shear matrix and diagonalizing.
Solution 1:
Look at the second column of the given matrix. The column is $v=(0,1)^T$ and this mean that $Av=v$ so $v$ is an eigenvector associated to the eigenvalue $1$.
Since this matrix isn't diagonalizable then the eigenspace relative to the eigenvalue $1$ has the dimension $1$ and it's spanned by $v$.