What are the Long-Term Trends of a Matrix in Algebra?

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Discussion Overview

The discussion revolves around finding long-term trends of a matrix in the context of market share dynamics between two competing soft drink companies, Cola A and Cola B. Participants explore the application of matrix algebra to model transitions in market shares over time, particularly focusing on how to compute these trends using transition matrices.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant questions the time frame for the transition probabilities, indicating that without knowing the duration these probabilities apply to, the question cannot be answered definitively.
  • Another participant provides a specific transition matrix and suggests that the initial market share can be represented as a vector, proposing a method to calculate the market share after a specified number of time periods.
  • A third participant introduces the concept of a steady-state vector and provides an equation to find it, suggesting that this approach may simplify the process of determining long-term trends.
  • There is a clarification regarding the notation used, with one participant asking if there was a typo in the representation of the matrices.
  • A later reply confirms that the notation used for the matrices is correct, maintaining the distinction between the vector and the transition matrix.

Areas of Agreement / Disagreement

Participants express differing views on the necessary conditions for applying the transition matrix and the method for finding the steady-state vector. There is no consensus on the best approach to take or the implications of the time frame for the transition probabilities.

Contextual Notes

Limitations include the ambiguity regarding the time frame for the transition probabilities and the assumptions about the application of the matrix over time. The discussion does not resolve how to handle these uncertainties.

pocketlint59
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hi, first-time user here

although this question deals with algebra, I feel that it pertains to matricies and probability more

basically, how do i find long term trends of a matrix with algebra?

ex. 2 competing soft drink companines cola A, and cola B presently have 1/3 and 2/3 shares of the market, respectively.

92% of cola A drinmiers will buy cola A again , the remainder switch to cola B
85% of Cola B drinkers will buy cola B again, and the remainder will switch brands

What is the market share after 2 weeks?
please help me and thanks in advance
 
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What is the market share after 2 weeks?
This question is impossible to answer. You haven't told us whether "92% of cola A drinmiers will buy cola A again , the remainder switch to cola B" applies every day, week, 2 seconds, 5 years, etc. If this information is for a 2 week period, then you just need to apply the matrix once. If it's for a 1 week period, you have to apply it twice, etc. so as you can tell, we can't answer the question unless you tell us how many times the matrix would apply, which depends on what length of time the given information pertains to. Anyways, the initial market share is a 1x2 vector (or 2x1, it doesn't matter too much), v = (1/3 2/3). Your matrix A will be:

0.92 .0.08
0.15 .0.85

After one time period, the market share will be represented by vA, after n time periods the market share will be represented by vAn. If your time period is x weeks (if your time period is 1 day, then x = 1/7), then n = 2/x.

To find the long-term trends, you need to compute the limit as n approaches infinity of vAn. You will need to diagonalize A to make this simple. Do you know where to go from here?
 
time = weekly basis

initial matrix
A B
[0.33 0.67]​
transition matrix
A B
A [0.92 0.08]
B [0.15 0.85]​
it's finding the steady-state vector using algebra

i hope this helps
 
Last edited:
The steady state vector is that vector v that satisfies Av = v. What I told you about finding the limit may have been wrong. It will be much easier to find the v if you look at it like this:

Av = v
Av = Iv
Av - Iv = 0
(A - I)v = 0

You'll see that A-I is a rather nice matrix, and it will be easy to see what v should be from there.
 
is V in place of b because of a typo?
 
No, v and A are the matrices I said they were in post 2.
 

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