Consider the matrix \begin{equation*} F= \begin{bmatrix} -13 & -8 & -4\\ 12 & 7 & 4\\ 24 & 16 & 7 \end{bmatrix} \end{equation*} of Example CPMS3, Example EMS3 and Example ESMS3. $F$'s eigenvalues and eigenspaces are \begin{align*} \eigensystem{F}{3}{\colvector{-\frac{1}{2}\\\frac{1}{2}\\1}}\\ \eigensystem{F}{-1}{\colvector{-\frac{2}{3}\\1\\0},\,\colvector{-\frac{1}{3}\\0\\1}} \end{align*}
Define the matrix $S$ to be the $3\times 3$ matrix whose columns are the three basis vectors in the eigenspaces for $F$, \begin{equation*} S= \begin{bmatrix} -\frac{1}{2} & -\frac{2}{3} & -\frac{1}{3}\\ \frac{1}{2} & 1 & 0\\ 1 & 0 & 1 \end{bmatrix} \end{equation*}
Check that $S$ is nonsingular (row-reduces to the identity matrix, Theorem NMRRI or has a nonzero determinant, Theorem SMZD). Then the three columns of $S$ are a linearly independent set (Theorem NMLIC). By Theorem DC we now know that $F$ is diagonalizable. Furthermore, the construction in the proof of Theorem DC tells us that if we apply the matrix $S$ to $F$ in a similarity transformation, the result will be a diagonal matrix with the eigenvalues of $F$ on the diagonal. The eigenvalues appear on the diagonal of the matrix in the same order as the eigenvectors appear in $S$. So, \begin{align*} \similar{F}{S}&= \inverse{ \begin{bmatrix} -\frac{1}{2} & -\frac{2}{3} & -\frac{1}{3}\\ \frac{1}{2} & 1 & 0\\ 1 & 0 & 1 \end{bmatrix} } \begin{bmatrix} -13 & -8 & -4\\ 12 & 7 & 4\\ 24 & 16 & 7 \end{bmatrix} \begin{bmatrix} -\frac{1}{2} & -\frac{2}{3} & -\frac{1}{3}\\ \frac{1}{2} & 1 & 0\\ 1 & 0 & 1 \end{bmatrix}\\ &= \begin{bmatrix} 6 & 4 & 2\\ -3 & -1 & -1\\ -6 & -4 & -1 \end{bmatrix} \begin{bmatrix} -13 & -8 & -4\\ 12 & 7 & 4\\ 24 & 16 & 7 \end{bmatrix} \begin{bmatrix} -\frac{1}{2} & -\frac{2}{3} & -\frac{1}{3}\\ \frac{1}{2} & 1 & 0\\ 1 & 0 & 1 \end{bmatrix}\\ &= \begin{bmatrix} 3 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{bmatrix} \end{align*}
Note that the above computations can be viewed two ways. The proof of Theorem DC tells us that the four matrices ($F$, $S$, $\inverse{F}$ and the diagonal matrix) will interact the way we have written the equation. Or as an example, we can actually perform the computations to verify what the theorem predicts.