Is an eigenvector of ? If so, what is the eigenvalue?
Yes, the required given value is .
Eigenvalue:An eigenvalue of A is a scalar such that the equation has a nontrivial solution.
Consider a invertible matrix A and is an eigenvector of the matrix A with respect to the eigenvalue .
The objective is to find whether the vector is eigenvector for is or not. If yes find the eigenvalue.
Since is an eigenvector of the matrix A with respect to the eigenvalue .
Find whether the vector is eigenvector for is or not as,
Therefore, from equation (1) the vector is eigenvector for is with respect to the eigenvalue .
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.
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