The process of diagonalizing a matrix $A$ goes in two steps. First we find the eigenvalues, and then we find the eigenvectors. To find the eigenvalues we look at the characteristic polynomial of $A$, denoted $p_A(\lambda)$. This is an $n$-th order polynomial in a variable $\lambda$ defined by $$ p_A(\lambda) = \det(\lambda I - A).$$ For reasons explained in the following video, the eigenvalues of $A$ are precisely the roots of $p_A(\lambda)$.

We find the eigenvectors of $A$ one eigenvalue at a time. As explained in the following video, the eigenspace $E_\lambda$ is the same as the null space of $A-\lambda I$, or equivalently the null space of $\lambda I - A$. A basis for this space is obtained by row reducing $\pm(A-\lambda I)$, exactly as in Appendix A.