Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 ∈ $\mathbb{Z}$.

share|cite|improve this question

2 Answers 2

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

share|cite|improve this answer

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

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.