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?