In Exercises 24 through 29, consider a dynamical system x → ( t + 1 ) = A x → ( t ) with two components. The accompanying sketch shows the initial state vector x → 0 and two eigenvectors, υ → 1 and υ → 2 , of A (with eigenvalues λ 1 and λ 2 , respectively). For the given values of λ 1 and λ 2 , sketch a rough trajectory. Consider the future and the past of the system. 28. λ 1 = 1.2 , λ 2 = 1.1
In Exercises 24 through 29, consider a dynamical system x → ( t + 1 ) = A x → ( t ) with two components. The accompanying sketch shows the initial state vector x → 0 and two eigenvectors, υ → 1 and υ → 2 , of A (with eigenvalues λ 1 and λ 2 , respectively). For the given values of λ 1 and λ 2 , sketch a rough trajectory. Consider the future and the past of the system. 28. λ 1 = 1.2 , λ 2 = 1.1
Solution Summary: The author illustrates the rough trajectory of the system for the Eigen values.
In Exercises 24 through 29, consider a dynamical system
x
→
(
t
+
1
)
=
A
x
→
(
t
)
with two components. The accompanying sketch shows the initial state vector
x
→
0
and two eigenvectors,
υ
→
1
and
υ
→
2
, of A (with eigenvalues
λ
1
and
λ
2
, respectively). For the given values of
λ
1
and
λ
2
, sketch a rough trajectory. Consider the future and the past of the system.
28.
λ
1
=
1.2
,
λ
2
=
1.1
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
A) Find the Eigenvalue and Eigenvector
B) Classify the critical point (0,0) and state
if it's stable or unstable.
c) Sketch the trajectories in a phase plane.
dx
3
-4
X
dt
1
-1
Suppose that i = Ax. Draw the phase portrait where the 2 x 2 matrix A has
complex eigenvalue A = +5i with i 0.
1. Consider the linear system
where
2
[³]
* = Ax₂_x(0) = [
x
A=[
-3 2
-4 1
a. Find the complex eigenvalues and eigenvectors of A
b. Sketch the phase portrait and classify the critical point at the origin
as a spiral point or center and determine its stability
c. Solve the system.
Chapter 7 Solutions
Linear Algebra With Applications (classic Version)
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