I'm studying the module Algebra, and was reading the course book "Álgebra. Aplicaciones a Teoría de Códigos". I got stuck on this problem because I couldn't prove a statement.
Question: A laboratory studies a species of bacteria that occupies the zones A and B of a tissue. Initially half of them are in A and the other half in B, but every minute, 20$\%$ of those of A pass to B and 30$\%$ of B pass to A. This behavior persists for several days. The distribution of bacteria at the minute $k$ can be represented by the vector $v_k=\begin{pmatrix} x_k \\ y_k \end{pmatrix}$, where $x_k$ and $y_k$ are the ratio (in so many per one) of bacteria in A and B respectively.
Prove that $\lim_{k \to \infty} (v_k) = \begin{pmatrix}
0.6x_k \\
0.4y_k
\end{pmatrix}$
My attempt:
We know that the recursive function for $v_k$ is $v_{k}=\begin{pmatrix}
x_{k-1} + 0.3y_{k-1} – 0.2x_{k-1} \\
y_{k-1} – 0.3y_{k-1} + 0.2x_{k-1}
\end{pmatrix} = \begin{pmatrix}
0.8x_{k-1}+0.3y_{k-1} \\
0.7y_{k-1}+0.2x_{k-1}
\end{pmatrix}$
We also know that the explicit function for $v_k$ is:
$v_k=\begin{pmatrix}
x_1+0.8x_{k} \\
y_1+0.7y_{k}
\end{pmatrix}$
Let us prove the statement.
We shall call $v_1, v_2, v_3, \cdots, v_k$ to the set of vectors of the study.
The sequence will be as such:
$v_1= \begin{pmatrix}
x \\
y
\end{pmatrix}$
$v_2= \begin{pmatrix}
0.8x+0.3y \\
0.7y+0.2x
\end{pmatrix}$
$v_3= \begin{pmatrix}
(0.8x+0.3y)\cdot 0.8 + (0.7y+0.2x) \cdot 0.3 \\
(0.7y+0.2x)\cdot 0.7 + (0.8x+0.3y) \cdot 0.2
\end{pmatrix} = \begin{pmatrix}
0.45x+0.7y \\
0.55y+0.3x
\end{pmatrix} = \begin{pmatrix}
0.75x \\
1.25y
\end{pmatrix}
$
However, already at $k=3$ we can see that the ratio $\begin{pmatrix}
0.6 \\
0.4
\end{pmatrix}$ is not being accomplished.
Best Answer
Let $A:=\begin{pmatrix} 8/10&3/10 \\ 2/10&7/10 \end{pmatrix} $, so by your formula $v_{k}=A\cdot v_{k-1}$.
Since linear map are continuous, it follows $$v=\lim_{k\to \infty} v_k= \lim_{k\to \infty} A\cdot v_{k-1}= A\cdot ( \lim_{k\to \infty}v_{k-1})= A\cdot v$$ In other words, the limit $v$ (if it exists, but I assume you have already proven it) is an eigenvector of $A$ with eigenvalue $1$.
The characteristic polynomial of $A$ is indeed $(\lambda-1)(\lambda-\frac 12)$ with eigenspac $V_1=\left< (3,2)\right>$ (the other is $V_{\frac 12}=\left< (-1,1)\right>$) so the ratio is precisely $\frac 32= \frac{0.6}{0.4}$.