The Action of a Linear Operator and an Introduction to Eigenvalues & Eigenvectors

Published on
07/20/2005

Activity Overview

In this Derive™ activity, students use matrix multiplication as a tool for drawing the outlines of a simple stick house. They work in two dimensional space and show the results of multiplying a figure by 2 X 2 matrices.

Before the Activity

See the attached DFW file for detailed instructions for this activity

Print pages from the attached DFW file for your class

During the Activity

Distribute the pages to the class.

Follow the Activity Procedures:

Set up a 2D plot window

Initialize the state of Derive's random number generator by entering a random number

Generate a 2 X 2 random number matrix

Multiply the number with the matrix and save the resultant matrix as variable A

Define a matrix H, of order, say, 10 X 2

Run the ShowAction Derive function to obtain the transpose of matrix H

Multiply the transpose of H on the left by A

Examine the basic shape of the house (It is squashed and the bottom angles are not right angles)

Define a matrix B to retain properties of the house, except its window orientation

Run the ShowAction Derive function and note that the new house is the same house as the original one, but rotated

Realize that the rotation is due to the trigonometric elements in the rotation matrix B

Clear the plot window of plots, redraw the house using matrix A

Run the ShowAction Derive function for successive powers

Understand that the action of higher powers of A appears to be collapsing the house to a straight line

Calculate the Eigen value λ for matrix A

Check whether the determinant of A - λI = 0

After the Activity

Students answer questions listed on the activity sheet.

Review student results:

As a class, discuss questions that appeared to be more challenging