Maple Exercises:
Instructions for the Maple Exercises:
- Staple these Maple Exercises to the end of all your handwritten work.
- Only the exercises shown in boldface are to be handed-in. All others
are strongly recommended but not to be handed-in.
- Write explanations in your Maple work, describing what you are doing and why. Don't just
hand-in a string of Maple computations with no explanation. The explanations can be
written-in by hand, or by using the "T" ("Text") button in Maple.
- Clearly label each Maple problem.
The Maple Exercises:
[See the posted Eigenvalues and
Eigenvectors, the posted Linear
Transformations and Eigenvectors, and the posted Diagonalization Maple Worksheets.]
- jh1: In exercises jh1 and jh2 of Homework Assignment #5,
you were asked to construct matrices which are a 5th root and a 4th root of the 5 by 5 identity
matrix. What can you deduce abut the eigenvalues of these matrices? Test your deductions
by finding the eigenvalues of these matrices. (See the posted Solutions
to Assignment 5 - Maple.) Find the modulus of each of the eigenvalues using "abs".
Plot the eigenvalues using "complexplot". Explain.
- jh2: In exercise jh3 of Homework Assignment #5,
you were asked to construct a matrix which is a 5th root of the 5 by 5 zero
matrix. What can you deduce abut the eigenvalues of this matrix? Test your deduction
by finding the eigenvalues of this matrix. (See the posted Solutions
to Assignment 5 - Maple.)
- jh3: In exercise jh4 of Homework Assignment #5,
you were asked to construct a 3 by 3 matrix of rank 1 which is its own square.
What can you deduce abut the eigenvalues of this matrix? Test your deduction
by finding the eigenvalues of this matrix. (See the posted Solutions
to Assignment 5 - Maple.)
- jh4: What are the eigenvalues of the 2 by 2 identity matrix? What is the
corresponding eigenspace? Plot using "eigenvectorplot" from Linear
Transformations and Eigenvectors. Explain.
- jh5: Using the method of exercise jh1 in Homework
Assignment #5, construct a 2 by 2 matrix which is a square root of the 2 by 2
identity matrix. Find its eigenvalues and eigenvectors. Plot using "eigenvectorplot"
from Linear Transformations and Eigenvectors. Repeat
for two more of these matrices constructed by re-executing the commands. (The square
root of a 2 by 2 matrix is built starting from a random 2 by 2 matrix.) Are the results
the same? Explain.
- jh6: The 2 by 2 rotation matrix is defined by equation 3, section 2.3, page 150
of the textbook. Construct the 2 by 2 rotation matrix for a rotation in the counterclockwise
direction by an angle of 30 degrees. Plot using "eigenvectorplot" from
Linear Transformations and Eigenvectors. Explain.
- jh7: As shown in the posted Diagonalization Maple
Worksheet, a symmetric matrix is real diagonalizable. Construct a 10 by 10 symmetric
matrix whose elements are random integers from 0 to 1. Swap the first two columns.
Is the matrix still real diagonalizable?
- jh8: Construct a 9 by 9 matrix whose eigenvalues are the 9 digits
of your student ID number, and whose eigenvectors are the column vectors of a 9 by 9
non-singular matrix whose elements are "random" integers from -1 to 1. (If the "random"
integers give a singular matrix, try again until obtaining a non-singular matrix.)
Check by finding the eigenvalues and eigenvectors of your matrix. What happens for
repeated digits in your student ID number?