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Unformatted text preview: Notes on matrices and discrete systems Patrick De Leenheer * September 3, 2007 Abstract These notes complement section 9 . 1 to 9 . 3 in our Nagle/Saff/Snider text, and illustrate the different concepts from linear algebra described there in the context of discrete time linear systems. If you have no, or limited background in linear algebra, you should carefully review both these notes and the sections from the book. 1 A motivating example Let’s consider a model of some population with 2 stages, a juvenile and an adult stage. Let J ( t ), M ( t ) denote the number of juveniles and adults at time t . Since we only take a census of the population every year (or month, or day), time is discrete here, so t = 0 , 1 , 2 , 3 ,... We will assume that juveniles turn into adults and that adults generate offspring. Our first goal is to make a precise mathematical model that describes this process. Secondly, we would like to use that model to make predictions about the future composition of the population given the initial composition. Assumptions for model : 1. We assume that none of the juveniles or adults die. 2. Let f be the fraction of juveniles that matures to adulthood by the next census. Then 1 f is the fraction of juveniles that remains in the juvenile class. 3. Once an individual becomes an adult, it remains an adult. 4. Let m be the number of offspring generated by 1 adult in 1 unit of time. Of course, we could immediately criticize this model. For instance, it is quite silly to assume that no individual ever dies. For now, we will ignore this, and later we will actually modify our model to reflect death. Question : Given the current composition of the population, what is the composition at the time of the next census? In other words, knowing J ( t ) and M ( t ), what will J ( t + 1) and M ( t + 1) be? Well, fJ ( t ) of juveniles grow into adults and all M ( t ) adults remain adult, so M ( t + 1) = fJ ( t ) + M ( t ) . Similarly, (1 f ) J ( t ) juveniles remain juveniles while mM ( t ) offspring are born which enter the juvenile class, so J ( t + 1) = (1 f ) J ( t ) + mM ( t ) So our model, which holds for all t = 0 , 1 , 2 ,... is: J ( t + 1) = (1 f ) J ( t ) + mM ( t ) M ( t + 1) = fJ ( t ) + M ( t ) But what does all this have to do with matrices? Well, it turns out that it is very useful to compactify the above model by defining certain vectors (which is just a column of numbers) and a matrix (which is really just a table of numbers). Let x ( t ) = J ( t + 1) M ( t + 1) and M = 1 f m f 1 * Email: [email protected] Department of Mathematics, University of Florida. 1 Then x ( t ) is a twodimensional vector and M is a 2 × 2 matrix. Formally, x ( t ) ∈ R 2 and A ∈ R 2 × 2 ....
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 Summer '06
 DeLeenheer
 Linear Algebra, Matrices, CN

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