Skip to main content

Authors in a Markov matrix: Which author do people find most inspiring? (19)

In the last article, we saw the number of inhabitants became one specific value. I raised a question, does this result depend on the initial state? In other words, are these conversing numbers the property of the matrix only or the property of both the matrix and the initial population vector?

  • Hypothesis 3: If the initial state is different, the result may change. For example, we started Berlin 900, Potsdam 100. If we started Berlin 0, Potsdam 1000, the result maybe not the same.

We can compute this by octave again. Set the initial state Berlin 0, Potsdam 1000.


octave:10> p = [0 1000]';
octave:11> M * p
   300
   700
octave:12> M^2 * p
   450.00
   550.00
octave:13> M^3 * p
   525.00
   475.00
octave:14> M^10 * p
   599.41
   400.59
octave:15> M^100 * p
   600.00
   400.00


Surprisingly, the results are the same even if we changed the initial state.  Figure 12 shows many different initial states and how they are changed after many years. I believe you see a pattern. And finding a pattern is mathematics.
Figure 11: Population history with various initial conditions.
By the way, do you realize the inhabitants Berlin 600 and Potsdam 400 are the special numbers? Let's compute how many people are moving in this number.
\begin{eqnarray*} \mbox{Berlin} \rightarrow \mbox{Potsdam} &=& 600 * 0.2 \\ &=& 120 \\ \mbox{Potsdam} \rightarrow \mbox{Berlin} &=& 400 * 0.3\\ &=& 120 \end{eqnarray*}
When once the number of inhabitants becomes this number, the number of moving people from Berlin to Potsdam and from Potsdam to Berlin become the same. Therefore, once the inhabitants of the cities became this number, the number of inhabitants will not change anymore. As you see in Figure 11, there is also something that doesn't depend on the initial number of total population. This suggests there is some kind of property in the matrix, not in the population vector. Mathematics first thinks the mathematical object, namely, numbers. Initially, Mathematics's main concern was how we could operate the numbers, i.e., addition, subtraction, and so on. But one point, the concern of mathematics was moved to the operation. The mathematicians stopped thinking about numbers. Here, how many inhabitants was the first concern, however, the next level becomes what the matrix is.

I would like to talk about that mathematicians stopped thinking about numbers in the next article. It would be related with mathematics, but not so much.

Comments

Popular posts from this blog

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix.  Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume  x  where x is in the null space of A .  The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the null spa

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides:  A bit intuitive (for me) explanation of area of parallelogram  (to my site, external link) .