4.1: Problem Set
EXERCISE \(\PageIndex{1}\)
Suppose \(\Lambda\) is \(n \times n\) matrix and T is a \(n \times n\) invertible matrix. use mathematical induction to show that:
\((T^{-1}\Lambda T)^k = T^{-1}\Lambda^{k}T\),
for all natural numbers k, i.e., \(k = 1, 2, 3, \dots\)
EXERCISE \(\PageIndex{2}\)
Suppose A is a \(n \times n\) matrix. Use the exponential series to give an argument that:
\(\frac{d}{dt}e^{At} = Ae^{At}\).
(You are allowed to use \(e^{A(t+h)} = e^{At}e^{Ah}\) without proof, as well as the fact that A and \(e^{At}\) commute, without proof.)
EXERCISE \(\PageIndex{3}\)
Consider the following linear autonomous vector field:
\(\dot{x} = Ax , x(0) = x_{0} , x \in \mathbb{R}^n\),
where A is a \(n \times n\) matrix of real numbers.
- Show that the solutions of this vector field exist for all time.
- Show that the solutions are infinitely differentiable with respect to the initial condition, \(x_{0}\).
EXERCISE \(\PageIndex{4}\)
Consider the following linear autonomous vector field on the plane:
\(\begin{pmatrix} {\dot{x_{1}}}\\ {\dot{x_{2}}} \end{pmatrix} = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} \begin{pmatrix} {x_{1}}\\ {x_{2}} \end{pmatrix}\)
(a) Describe the invariant sets.
(b) Sketch the phase portrait.
(c) Is the origin stable or unstable? Why?
EXERCISE \(\PageIndex{5}\)
Consider the following linear autonomous vector field on the plane:
\(\begin{pmatrix} {\dot{x}}\\ {\dot{y}} \end{pmatrix} = \begin{pmatrix} {0}&{0}\\ {0}&{0} \end{pmatrix} \begin{pmatrix} {x_{1}}\\ {x_{2}} \end{pmatrix}\)
(a) Describe the invariant sets.
(b) Sketch the phase portrait.
(c) Is the origin stable or unstable? Why?