Download original notes
PAGE 1

5.2 (continued)

Example

\[ \vec{x}' = \begin{bmatrix} -1 & 2 \\ -2 & -1 \end{bmatrix} \vec{x} \]

Eigenvalues:

\[ \begin{vmatrix} -1-\lambda & 2 \\ -2 & -1-\lambda \end{vmatrix} = 0 \]

Solving the characteristic equation:

\[ (-1-\lambda)^2 + 4 = 0 \]\[ (-1-\lambda)^2 = -4 \]\[ (-1-\lambda) = \pm 2i \]

Complex conjugate pairs

\[ \lambda = -1 \pm 2i \]

Eigenvectors: \( (A - \lambda I) \vec{v} = \vec{0} \)

For \( \lambda = -1 + 2i \):

\[ \begin{bmatrix} -2i & 2 & 0 \\ -2 & -2i & 0 \end{bmatrix} \]
\( \rightarrow \)
\[ \begin{bmatrix} -2 & -2i & 0 \\ -2i & 2 & 0 \end{bmatrix} \]

multiply row 1 by \( -i \)

add to row 2

PAGE 2

Eigenvalue and Eigenvector Calculations

\[ \rightarrow \begin{bmatrix} -2 & -2i & 0 \\ 0 & 0 & 0 \end{bmatrix} \]\[ \rightarrow \begin{bmatrix} 1 & i & 0 \\ 0 & 0 & 0 \end{bmatrix} \quad \vec{v} = \begin{bmatrix} -i \\ 1 \end{bmatrix} \]

Repeat with \( \lambda = -1 - 2i \)

\[ \begin{bmatrix} 2i & 2 & 0 \\ -2 & 2i & 0 \end{bmatrix} \rightarrow \dots \rightarrow \begin{bmatrix} 1 & -i & 0 \\ 0 & 0 & 0 \end{bmatrix} \]\[ \vec{v} = \begin{bmatrix} i \\ 1 \end{bmatrix} \]
eigenvectors are conjugate pairs

Solutions: \( e^{\lambda t} \vec{v} \)

\[ \lambda = -1 + 2i, \quad \vec{v} = \begin{bmatrix} -i \\ 1 \end{bmatrix} \]\[ \lambda = -1 - 2i, \quad \vec{v} = \begin{bmatrix} i \\ 1 \end{bmatrix} \]

Using the top pair:

\[ e^{(-1 + 2i)t} \begin{bmatrix} -i \\ 1 \end{bmatrix} \]
PAGE 3

Real-Valued Solutions for Complex Eigenvalues

We want the real-valued solutions to not contain \( i \).

\[ e^{-t} e^{i(2t)} \begin{bmatrix} -i \\ 1 \end{bmatrix} \]

Euler's identity

\[ e^{it} = \cos(t) + i \sin(t) \]
\[ = e^{-t} (\cos(2t) + i \sin(2t)) \begin{bmatrix} -i \\ 1 \end{bmatrix} \]
\[ = e^{-t} \begin{bmatrix} \sin(2t) - i \cos(2t) \\ \cos(2t) + i \sin(2t) \end{bmatrix} \]
\[ = \underbrace{e^{-t} \begin{bmatrix} \sin(2t) \\ \cos(2t) \end{bmatrix}}_{\text{real part}} + i \underbrace{e^{-t} \begin{bmatrix} -\cos(2t) \\ \sin(2t) \end{bmatrix}}_{\text{imaginary part}} \]

Using the Conjugate Pair

Using the other pair, we get:

\[ e^{\lambda t} \vec{v} = e^{-t} \begin{bmatrix} \sin(2t) \\ \cos(2t) \end{bmatrix} - i e^{-t} \begin{bmatrix} -\cos(2t) \\ \sin(2t) \end{bmatrix} \]
Solutions are conjugate pairs
PAGE 4

General Solution and Phase Behavior of Linear Systems

The real and imaginary parts of either solution are themselves solutions to \(\vec{x}' = A\vec{x}\). So, we use them as fundamental solutions to form the general solution.

\[ \vec{x} = c_1 e^{-t} \begin{bmatrix} \sin(2t) \\ \cos(2t) \end{bmatrix} + c_2 e^{-t} \begin{bmatrix} -\cos(2t) \\ \sin(2t) \end{bmatrix} \]

Phase Diagram: Periodic Behavior

  • Spirals ( into origin if real part of \(\lambda\) is \(< 0\); away from origin if real part of \(\lambda\) is \(> 0\) )
  • If real part of \(\lambda\) is \(0 \rightarrow\) ovals

Spiral Direction? Clockwise / Counterclockwise?

Easy way: pick convenient \(\vec{x}\) in \(\vec{x}' = A\vec{x}\).

\[ \vec{x}' = \begin{bmatrix} -1 & 2 \\ -2 & -1 \end{bmatrix} \vec{x} \]

tangent vectors

Pick \(\vec{x} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\)

\[ \vec{x}' = \begin{bmatrix} -1 \\ -2 \end{bmatrix} \]

left and down → clockwise

PAGE 5
A phase portrait in a Cartesian coordinate system with axes labeled x1 and x2. The diagram shows a spiral sink where trajectories spiral inwards towards the origin in a clockwise direction. Several spiral paths are shown, each with arrows indicating the direction of flow towards the center (0,0).
Figure 1: Phase portrait showing a spiral sink in the \(x_1-x_2\) plane.

5.5 Repeated Eigenvalues

\[ \vec{x}' = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \vec{x} \quad \lambda = 1, 1 \quad \text{algebraic multiplicity is Two} \]

eigenvectors: \((A - \lambda I)\vec{v} = \vec{0}\)

\[ \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \vec{v} = \begin{bmatrix} a \\ b \end{bmatrix} \]

\[ = a \begin{bmatrix} 1 \\ 0 \end{bmatrix} + b \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]

two linearly indp vectors we use as eigenvectors

PAGE 6

Linear Systems and Geometric Multiplicity

\(\vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)

geometric multiplicity is Two

two solutions \(e^{\lambda t} \vec{v}\): \(e^t \begin{bmatrix} 1 \\ 0 \end{bmatrix}, e^t \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)

general solution: \(\vec{x} = c_1 e^t \begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_2 e^t \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)

Phase Diagram

A phase diagram in a Cartesian coordinate system with axes labeled x_1 and x_2. The diagram shows a star-shaped pattern of trajectories (phase lines) all radiating outward from the origin (0,0). Every line has an arrow pointing away from the center, indicating an unstable node where solutions grow exponentially over time in all directions.

Defective Matrices

now let's look at \(\vec{x}' = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix} \vec{x}\) with \(\lambda = 2, 2\)

alg. mult. is two

\((A - \lambda I) \vec{v} = \vec{0}\)

\(\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \quad \vec{v} = \begin{bmatrix} a \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\)

geo. mult. is one

missing one vector

(matrix A is defective, defect of one)

PAGE 7

Solutions for Systems with Repeated Eigenvalues

Solution 1: \( e^{\lambda t} \vec{v} \)

Solution 2: \( e^{\lambda t} (t \vec{v} + \vec{u}) \)

\( \vec{u} \) is a generalized eigenvector

where \( (A - \lambda I) \vec{u} = \vec{v} \) and \( \vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)

here, \( \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)
\( \vec{u} = \begin{bmatrix} a \\ 1 \end{bmatrix} \)
Choose any \( a \) as long as \( \vec{u} \neq \vec{0} \) and is linearly independent from \( \vec{v} \)

here, \( a = 0 \) and \( \vec{u} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)

General Solution:

\[ \vec{x} = c_1 e^{2t} \begin{bmatrix} 1 \\ 0 \end{bmatrix} + c_2 e^{2t} \left( t \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) \]