In Otto & Day's (2007) A Biologist's Guide to Mathematical Modelling, they give the general solution to discrete-time linear models as:
$\vec{n(t)} = \boldsymbol{A D^t A^{-1}}\vec{n(0)}$
where $\vec{n(t)}$ is a vector of solutions at time $t$ for the $d$ variables in the system, $\boldsymbol{A}$ is a $d \times d$ matrix of column-wise right eigenvectors, and $D^t$ is a $d \times d$ diagonal matrix of eigenvalues.
For the system with matrix of coefficients $\boldsymbol{M} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the complex eigenvalues are:
$\vec{\lambda} = \frac{a + d \pm \sqrt{ - (a-d)^2 - 4bc}}{2}$
which can be written as $\lambda_{1} = \alpha + \beta i = R(\cos(\beta t) + i \sin(\beta t))$ giving:
$\boldsymbol{D}^t = \begin{bmatrix} R(\cos(\theta t) + i\sin(\theta t)) & 0 \\ \ & R(\cos(\theta t) - i \sin(\theta t)) \end{bmatrix}$
The matrix $\boldsymbol{A}$ is:
$ \boldsymbol{A} = \begin{bmatrix} 1 & 1 \\ -a + d + \sqrt{-(a-d)^2 - 4bc} & -a + d - \sqrt{-(a-d)^2 - 4bc} \end{bmatrix} $
Otto and Day multiply out $\boldsymbol{A D^t A^{-1}}$ to get:
$ R^t \begin{bmatrix} \cos(\theta t) + \frac{a-d}{2 \beta}sin(\theta t) & \frac{b}{\beta} \sin(\theta t) \\ \frac{c}{\beta}\sin(\theta t) & \cos(\theta t) - \frac{a-d}{2 \beta} \sin(\theta t) \end{bmatrix} $
My problem is that I am struggling to multiply out the matrices to get their result.
From the definitions of $\boldsymbol{A}$ and $\boldsymbol{D}^t$, I first derived $\boldsymbol{A}^{-1}$ as:
$ \boldsymbol{A^{-1}} = \begin{bmatrix} \frac{a - d + \sqrt{-(a-d)^2 - 4bc}}{2\sqrt{-(a-d)^2 - 4bc}} & \frac{1}{\sqrt{-(a-d}^2 - 4bc} \\ \frac{-a + d + \sqrt{-(a-d)^2- 4bc}}{2\sqrt{-(a-d)^2 - 4bc}} & \frac{1}{-\sqrt{-(a-d}^2 - 4bc} \end{bmatrix} $
I then find $\boldsymbol{AD}$ as:
$ \boldsymbol{AD} = \begin{bmatrix} R(\cos(\theta t) + i\sin(\theta t)) & R(\cos(\theta t) - i\sin(\theta t)) \\ \frac{(-a+d+\sqrt{-(a-d)^2 + 4bc})R(\cos(\theta t) + i\sin(\theta t)}{2} & \frac{(-a+d-\sqrt{-(a-d)^2 + 4bc})R(\cos(\theta t) - i\sin(\theta t)}{2} \end{bmatrix} $
But then multiplying by $\boldsymbol{A^{-1}}$ does not give me the desired answer. To save space, I'll take the first element of $\boldsymbol{A D^{1} A^{-1}}$:
$\boldsymbol{(A D^{1} A^{-1}})_{11} = R(\cos(\theta t) + i\sin(\theta t))\frac{a - d + \sqrt{-(a-d)^2 - 4bc}}{2\sqrt{-(a-d)^2 - 4bc}} + R(\cos(\theta t) - i\sin(\theta t))\frac{-a + d + \sqrt{-(a-d)^2- 4bc}}{2\sqrt{-(a-d)^2 - 4bc}}\\ = \frac{R}{\sqrt{-(a-d)^2 - 4bc}}\big[(\sqrt{(a-d)^2 - 4bc}) \cos(\theta t) + (i)\sin(\theta t)(a - d)\big] $
I am now unsure how they Otto and Day got the first element of their solution. Can anyone give me a push in the right direction?