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 2 years ago
Can one take the limit of a matrix as it is multiplied by itself infinite times? We are using a matrix to model how a population changes over a given period of time. Repeated multiplication of the matrix will show how the percentages change after each period of time transpires. The question was will the population stabilize. So I reasoned that if the limit of repeated multiplications of the matrix approached specific values, that the population would indeed stabilize.
 2 years ago
Can one take the limit of a matrix as it is multiplied by itself infinite times? We are using a matrix to model how a population changes over a given period of time. Repeated multiplication of the matrix will show how the percentages change after each period of time transpires. The question was will the population stabilize. So I reasoned that if the limit of repeated multiplications of the matrix approached specific values, that the population would indeed stabilize.

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Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1Yes, you can do that. Look up Markov Matrix.

SBurchette
 2 years ago
Best ResponseYou've already chosen the best response.0Example/ Matrix A represents how the population changes over a week. Matrix B represents the population. So\[\lim_{t \rightarrow \infty}(B)A^t\] Would represent the population after t numbers of weeks transpire. I took this limit numerically and the product matrix consistently approached particular values.

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1I'll explain.... are you familiar with diagonalization of a matrix?

SBurchette
 2 years ago
Best ResponseYou've already chosen the best response.0Not entirely, I'm in a elementary linear algebra class. We have just been covering matrix operations.

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1I see. I'm not quite sure how to explain it to you without referencing diagonalization or at least eigenvalues and eigenvectors, but you are correct in that if the rows and columns of the matrix add to 1 and all of the entries are positive, repeated multiplication by itself will approach a constant result

SBurchette
 2 years ago
Best ResponseYou've already chosen the best response.0Ok, I suppose the question more of an application than a presentation of theory. I do anticipate diving deeper into matrix theory =) Thanks for the assistance.

Jemurray3
 2 years ago
Best ResponseYou've already chosen the best response.1What you stumbled on is actually a very deep result in matrix theory, it's just that you're not quite prepared to appreciate it yet. You'll get there soon, though.
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