I'm wondering how one can generally go about to diagonalize a linear transformation, especially using a basis consisting of eigenvectors.
For example, this problem:
Let $T: \mathbb{R}^3 → \mathbb{R}^3$ be the linear operator which is represented in the standard ordered basis by the matrix
$$ \begin{bmatrix} -9 & 4 & 4 \\ -8 & 3 & 4 \\ -16 & 8 & 7 \end{bmatrix} $$
Prove that $T$ is diagonalizable by exhibiting a basis for $\mathbb{R}^3$ consisting of eigenvectors for $T$.