5.1: Problem Set

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EXERCISE \(\PageIndex{1}\)

Suppose A is a \(n \times n\) matrix of real numbers. Show that if \(\lambda\) is an eigenvalue of A with eigenvector e, then \(\bar{\lambda}\) is an eigenvalue of A with eigenvector \(\bar{e}\).

EXERCISE \(\PageIndex{2}\)

Consider the matrices:

\(A_{1} = \begin{pmatrix} {0}&{-w}\\ {w}&{0} \end{pmatrix}, A_{2} = \begin{pmatrix} {0}&{w}\\ {-w}&{0} \end{pmatrix}, w>0\)

Sketch the trajectories of the associated linear autonomous ordinary differential equations:

\(\begin{pmatrix} {\dot{x_{1}}}\\ {\dot{x_{2}}} \end{pmatrix} = A_{i}\begin{pmatrix} {x_{1}}\\ {x_{2}} \end{pmatrix}, i = 1, 2\)

EXERCISE \(\PageIndex{3}\)

Consider the matrices:

\(A = \begin{pmatrix} {-1}&{-1}\\ {-9}&{-1} \end{pmatrix}\)

(a) Show that the eigenvalues and eigenvectors are given by:

\(-1-3i: \begin{pmatrix} {1}\\ {3i} \end{pmatrix} = \begin{pmatrix} {1}\\ {0} \end{pmatrix}+i\begin{pmatrix} {0}\\ {3} \end{pmatrix}\)

\(-1+3i: \begin{pmatrix} {1}\\ {-3i} \end{pmatrix} = \begin{pmatrix} {1}\\ {0} \end{pmatrix}-i\begin{pmatrix} {0}\\ {3} \end{pmatrix}\)

(b) Consider the four matrices:

\(T_{1} = \begin{pmatrix} {1}&{0}\\ {0}&{-3} \end{pmatrix}\)

\(T_{2} = \begin{pmatrix} {1}&{0}\\ {0}&{3} \end{pmatrix}\)

\(T_{3} = \begin{pmatrix} {0}&{1}\\ {-3}&{0} \end{pmatrix}\)

\(T_{4} = \begin{pmatrix} {0}&{1}\\ {3}&{0} \end{pmatrix}\)

Compute \(\Lambda_{i} = T_{i}^{-1}AT_{i}, i = 1 \dots 4\).

EXERCISE \(\PageIndex{4}\)

Consider the following two dimensional linear autonomous vector field:

\(\begin{pmatrix} {\dot{x_{1}}}\\ {\dot{x_{2}}} \end{pmatrix} = \begin{pmatrix} {-2}&{1}\\ {-5}&{2} \end{pmatrix} \begin{pmatrix} {x_{1}}\\ {x_{2}} \end{pmatrix}, (x_{1}(0), x_{2}(0)) = (x_{10}, x_{20})\).

Show that the origin is Lyapunov stable. Compute and sketch the trajectories.

EXERCISE \(\PageIndex{5}\)

Consider the following two dimensional linear autonomous vector field:

\(\begin{pmatrix} {\dot{x_{1}}}\\ {\dot{x_{2}}} \end{pmatrix} = \begin{pmatrix} {1}&{2}\\ {2}&{1} \end{pmatrix} \begin{pmatrix} {x_{1}}\\ {x_{2}} \end{pmatrix}, (x_{1}(0), x_{2}(0)) = (x_{10}, x_{20})\).

Show that the origin is a saddle. Compute the stable and unstable subspaces of the origin in the original coordinates, i.e. the \(x_{1}-x_{2}\) coordinates. Sketch the trajectories in the phase plane.

EXERCISE \(\PageIndex{6}\)

Compute \(e^{A}\), where

\(A = \begin{pmatrix} {\lambda}&{1}\\ {0}&{\lambda} \end{pmatrix}\)

Hint. Write

\(A = \underbrace{\begin{pmatrix} {\lambda}&{0}\\ {0}&{\lambda} \end{pmatrix}}_{\equiv S}+\underbrace{\begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix}}_{\equiv N}\)

Then

\(A \equiv S+N\), and NS = SN.

Use the binomial expansion fo compute \((S + N)^n\), \(n \ge 1\),

\((S + N)^n = \sum_{k=0}^{n} \begin{pmatrix} {n}\\ {k} \end{pmatrix} S^{k}N^{n-k}\),

where

\(\begin{pmatrix} {n}\\ {k} \end{pmatrix} = \frac{n!}{k!(n-k)!}\)

and substitute the results into the exponential series.