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How do I go about writing

$(a x_0+by_0)\begin{bmatrix}q\\r\end{bmatrix} e$λ1t + $(cx_0+dy_0)\begin{bmatrix}s\\t\end{bmatrix}$eλ2t

in the form

$\begin{bmatrix} w&x\\ y&z \end{bmatrix}\begin{bmatrix}x_0\\y_0\end{bmatrix}$

Where a,b,c,d,q,r,s,t,λ12 ∈ $\mathbb{Z}$.

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up vote 1 down vote accepted

$w = aqe^{{\lambda_1}t} + cse^{{\lambda_2}t}$

$x = bqe^{{\lambda_1}t} + dse^{{\lambda_2}t}$

$y = are^{{\lambda_1}t} + cte^{{\lambda_2}t}$

$z = bre^{{\lambda_1}t} + dte^{{\lambda_2}t}$


To see why:

$(a x_0+by_0)\begin{bmatrix}q\\r\end{bmatrix} e^{\lambda_1t} + (cx_0+dy_0)\begin{bmatrix}s\\t\end{bmatrix}e^{\lambda_2t}$

$= [a\ b]\begin{bmatrix}x_0\\y_0\end{bmatrix}\begin{bmatrix}q\\r\end{bmatrix} e^{\lambda_1t} + [c\ d]\begin{bmatrix}x_0\\y_0\end{bmatrix}\begin{bmatrix}s\\t\end{bmatrix} e^{\lambda_1t}$

$= e^{\lambda_1t}\begin{bmatrix}q\\r\end{bmatrix} [a\ b]\begin{bmatrix}x_0\\y_0\end{bmatrix} + e^{\lambda_2t}\begin{bmatrix}s\\t\end{bmatrix} [c\ d]\begin{bmatrix}x_0\\y_0\end{bmatrix}$ (this step works because $[a\ b]\begin{bmatrix}x_0\\y_0\end{bmatrix}$ for example is just a simple number)

$= (e^{\lambda_1t}\begin{bmatrix}q\\r\end{bmatrix}[a\ b] + e^{\lambda_2t}\begin{bmatrix}s\\t\end{bmatrix}[c\ d])\begin{bmatrix}x_0\\y_0\end{bmatrix}$

So the matrix $\begin{bmatrix} w&x\\ y&z \end{bmatrix}$ you want is basically just $(e^{\lambda_1t}\begin{bmatrix}q\\r\end{bmatrix}[a\ b] + e^{\lambda_2t}\begin{bmatrix}s\\t\end{bmatrix}[c\ d])$.

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Let $c_1=e^{\lambda_1 t}$ and $c_2=e^\lambda_2 t$.

Write $$\eqalign{ (a x_0+by_0)\begin{bmatrix}q\\r\end{bmatrix} c_1 + (cx_0+dy_0)\begin{bmatrix}s\\t\end{bmatrix} c_2 &= \begin{bmatrix}q(a x_0+by_0)c_1\\r(a x_0+by_0)c_1\end{bmatrix} + \begin{bmatrix}s(cx_0+dy_0)c_2\\t(cx_0+dy_0)c_2\end{bmatrix} \cr &=\left[\matrix{ q(a x_0+by_0)c_1 + s(cx_0+dy_0)c_2\cr r(a x_0+by_0)c_1+ t(cx_0+dy_0)c_2 } \right]\cr &= \left[\matrix{ (qa c_1 +scc_2)x_0 + (qbc_1+sdc_2)y_0 \cr (rac_1 +tcc_2)x_0+ (rbc_1+tdc_2)y_0 } \right]\cr \cr &= \left[\matrix{ qa c_1 +scc_2 & qbc_1+sdc_2 \cr rac_1 +tcc_2& rbc_1+tdc_2 } \right] \left[\matrix{x_0\cr y_0}\right]. \cr } $$

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