All problems below refer to a dynamical system of the form
x(k+1) = Ax(k) x(0) given
which we know has solutions of the form
x(k) = Akx(0)
In each case below, what can you say about the long term behavior of the system, given the information provided?
(The Principal Axis Theorem and Diagonalization help here!)
1. The eigenvalues of A are found to be: .7, -.5, 1.2 with eigenvectors, respectively, of:(-1,1,0), (0,-1,2) and ( 1,3,2)
2. It is a Markov Chain. The eigenvalues of A are: -.8, 1.0, and .25. Eigenvectors, respectively, are(-2.3, 1.7, 0), (.5,.3,.2) (-.5,0, 1.7)
3. It is a population model. The eigenvalues are -.3, .2, .26 and .9. The eigenvectors, respectively, are:(-1, 3, .65, .33), (-3,1,2,0), ( 1,2,1,7) and (2,1,1, .5)
4. The eigenvalues are 2, .8, -.3, and -2. The eigenvectors are, respectively,(1, 3,2,1), ( -8,1,4,-4) ( 0,0, 1 , 3) and (7,1,1,5)