My Math Forum Markov Chain Discrete Dynamical System

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 May 8th, 2017, 05:32 PM #1 Newbie   Joined: Mar 2015 From: Penns Posts: 8 Thanks: 0 Markov Chain Discrete Dynamical System Hey guys, so my teacher told us to look at the book and find a very similar problem and to just do it. I don't typically ask for the answer but I just need help setting up this problem. A drug is administered @ 100 units of the drug into the blood. Every 10 minutes, 50% of the drug goes to bloodstream & 50% to liver During this same time period 75% of the drug in the liver goes to the bloodstream while 25% stays in the liver. Initially injection gets process started 100 units of the drug directly to blood stream(0 to liver). How to model this system given x sub k = amount in blood stream after 10 min time interval. y sub k = amount of drug in the liver after k 10 min time intervals have passed. Using matrix algebra: X(k+1) = A * X(k)[X is a vector] where X(k)= Matrix x sub k over y sub k. Then there is another matrix showing A = (2x2) matrix top 2 values are 1/2 & 3/4. Bottom 2 values are 1/2 and 1/4. Can any one tell me what the hell is going on? I think I can use the markov chain or the matrix algebra. I'm super lost and a step by step is appreciated.
 May 8th, 2017, 05:56 PM #2 Senior Member     Joined: Sep 2015 From: Southern California, USA Posts: 1,604 Thanks: 817 if I understand this what you have is $\begin{bmatrix}b_{n+1} \\l_{n+1}\end{bmatrix}=\begin{bmatrix}0.5 &0.75\\0.5 &0.25\end{bmatrix}\begin{bmatrix}b_n \\ l_n\end{bmatrix}$ $\begin{bmatrix}b_0 \\ l_0\end{bmatrix}=\begin{bmatrix}100 \\ 0\end{bmatrix}$
May 8th, 2017, 06:35 PM   #3
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 Originally Posted by romsek if I understand this what you have is $\begin{bmatrix}b_{n+1} \\l_{n+1}\end{bmatrix}=\begin{bmatrix}0.5 &0.75\\0.5 &0.25\end{bmatrix}\begin{bmatrix}b_n \\ l_n\end{bmatrix}$ $\begin{bmatrix}b_0 \\ l_0\end{bmatrix}=\begin{bmatrix}100 \\ 0\end{bmatrix}$

That is correct. It's a bit confusing saying find all the eigen values & bases for all eigenspaces of the matrix A. I get eigen-values and all. But it's confusing in the form of a word problem. I think you help me set it up tho.

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