27: a. Based on your answers in Exercises 24 and 25, find closed formulas for the components of the dynamical system
with initial value . Then do the same for the initial value . Sketch the two trajectories.
b. Consider the matrix
Using technology, compute some powers of the matrix A, say, A2, A5, A10, . . . . What do you observe? Diagonalize matrix A to prove your conjecture. (Do not use Theorem 2.3.11, which we have not proven
is an arbitrary positive transition matrix, what can you say about the powers At as t goes to infinity? Your result proves Theorem 2.3.11c for the special case of a positive transition matrix of size 2 × 2.
Transition matrix may refer to: The matrix associated with a change of basis for a vector space. Stochastic matrix, a square matrix used to describe the transitions of a Markov chain. State-transition matrix, a matrix whose product with the state vector at an initial time.
(a) Clearly we see that,
(b) We compute,
(c) We can see that for t>0 applies that,
Hence, the final answer is : (a)
(a) Give an example of a 3 × 3 matrix A with as many nonzero entries as possible such that both span() and span(,) are A-invariant subspaces of . See Exercise 65.
(b) Consider the linear space V of all 3 × 3 matrices A such that both span () and span (,) are A-invariant subspaces of . Describe the space V (the matrices in V “have a name”), and determine the dimension of V.
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