Question

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.

Answer 1

Yes, you can do that. Look up Markov Matrix.

Answer 2

Example/ 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.

Answer 3

I'll explain.... are you familiar with diagonalization of a matrix?

Answer 4

Not entirely, I'm in a elementary linear algebra class. We have just been covering matrix operations.

Answer 5

I 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

Answer 6

Ok, 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.

Answer 7

What 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.