Education Technology

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

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
  • Re-teach concepts as necessary