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Found in: Page 426

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# For the values of ${{\mathbit{\lambda }}}_{{\mathbf{1}}}{\mathbf{=}}{\mathbf{1}}$ and ${{\mathbit{\lambda }}}_{{\mathbf{1}}}{\mathbf{=}}{\mathbf{-}}{\mathbf{1}}$, sketch the trajectories for all nine initial values shown in the following figures. For each of the points, trace out both future and past of the system.

The trajectories for all nine initial values is

See the step by step solution

## Step 1: Draw the future and past of the system.

Consider the Eigen values ${\lambda }_{1}=1$ and ${\lambda }_{2}=-1$.

As ${\lambda }_{1}>0>{\lambda }_{2}$, the future and past of the system toward outside.

Draw the graph of ${E}_{1}$ and ${E}_{-1}$ as follows:

## Step 2: Draw the trajectory of the system.

If the Eigen values of the system are ${{\mathbit{\lambda }}}_{{\mathbf{1}}}$ and ${{\mathbit{\lambda }}}_{{\mathbf{2}}}$ such that ${{\mathbit{\lambda }}}_{{\mathbf{1}}}{\mathbf{>}}{\mathbf{0}}{\mathbf{>}}{{\mathbit{\lambda }}}_{{\mathbf{2}}}$ then the phase portraits of the system is

By the trajectory definition, draw the graph of the trajectory of the system as follows:

Hence the trajectories for all nine initial values and traces of the future and past of the system are sketched.