According to Eqs. (7.37) and (7.38), we need first to find the eigenvectors and eigenvalues of the \pmb A_c matrix. We take the eigenvectors to be

\begin{bmatrix} t_{11} \\ t_{21} \end{bmatrix}            and            \begin{bmatrix} t_{12} \\ t_{22} \end{bmatrix}

\pmb {TA} = \pmb{FT} \\ \begin{bmatrix} \pmb t_1 & \pmb t_2 & \pmb t_3 \end{bmatrix} \begin{bmatrix} p_1 & 0 & 0 \\ 0 & p_2 & 0 \\ 0 & 0 & p_3 \end{bmatrix} = \pmb F \begin{bmatrix} \pmb t_1 & \pmb t_2 & \pmb t_3 \end{bmatrix} .                  (7.37)

p_i \pmb t_i = \pmb {Ft}_i, \ \ \ \ i = 1, 2, 3.                  (7.38)

The equations using the eigenvector on the left are

\begin{bmatrix} -7 & -12 \\ 1 & 0 \end{bmatrix} \begin{bmatrix}t_{11} \\ t_{21} \end{bmatrix} = p \begin{bmatrix} t_{11} \\ t_{21} \end{bmatrix} ,                  (7.39a)

−7t_{11} − 12t_{21} = pt_{11},                          (7.39b)

t_{11} = pt_{21}.                               (7.39c)

Substituting Eq. (7.39c) into Eq. (7.39b) results in

−7pt_{21} − 12t_{21} = p^2t_{21},                         (7.40a)

p^2t_{21} + 7pt_{21} + 12t_{21} = 0,                       (7.40b)

p^2 + 7p + 12 = 0,                            (7.40c)

p = −3, −4.                               (7.40d)

We have found (again!) that the eigenvalues (poles) are −3 and −4 ; furthermore, Eq. (7.39c) tells us that the two eigenvectors are

\begin{bmatrix} -4t_{21} \\ t_{21} \end{bmatrix}                 and                  \begin{bmatrix} -3t_{22} \\ t_{22} \end{bmatrix} ,

where t_{21} and t_{22} are arbitrary nonzero scale factors. We want to select the two scale factors such that both elements of \pmb B_m in Eq. (7.17a) are unity. The equation for \pmb B_m in terms of \pmb B_c is \pmb {TB}_m = \pmb B_c, and its solution is t_{21} = −1 and t_{22} = 1. Therefore, the transformation matrix and its inverse ^5 are

\pmb T = \begin{bmatrix} 4 & -3 \\ -1 & 1 \end{bmatrix}, \ \ \ \ \pmb T^{-1} = \begin{bmatrix} 1 & 3 \\ 1 & 4 \end{bmatrix}.                      (7.41)

Elementary matrix multiplication shows that, using T as defined by Eq. (7.41), the matrices of Eqs. (7.15) and (7.17) are related as follows:

\pmb A_c = \begin{bmatrix} -7 & -12 \\ 1 & 0 \end{bmatrix}, \ \ \ \ \pmb B_c = \begin{bmatrix}1 \\0 \end{bmatrix} ,            (7.15a)

  \pmb C_c = \begin{bmatrix} 1 & 2 \end{bmatrix} , \ \ \ \ D_c =0,              (7.15b)

\pmb A_m = \begin{bmatrix} -4 & 0 \\ 0 & -3 \end{bmatrix}, \ \ \ \ \pmb B_m = \begin{bmatrix} 1 \\ 1 \end{bmatrix} ,         (7.17a)

\pmb C_m = \begin{bmatrix} 2 & -1 \end{bmatrix} , \ \ \ \ D_m=0,                       (7.17b)

\pmb A_m = \pmb T^{−1} \pmb A_c \pmb T, \ \ \ \ \ \pmb B_m = \pmb T^{−1} \pmb B_c, \\ \pmb C_m = \pmb C_c \pmb T, \ \ \ \ \ \ D_m = D_c.                     (7.42)

These computations can be carried out by using the following MATLAB statements