Orthogonality
Part 4: Orthogonal matrices
An n x n matrix A is orthogonal
if its columns form an orthonormal set, i.e., if the columns of A form
an orthonormal basis for Rn.
- We construct an orthogonal
matrix in the following way. First, construct four random 4-vectors, v1,
v2, v3, v4. Then
apply the Gram-Schmidt process to these vectors to form an orthogonal set
of vectors. Then normalize each vector in the set, and make these vectors
the columns of A.
- Calculate ATA
and AAT. What do you conclude from the results?
- Construct random 4-vectors
x and y. Then compute the dot products of x and y
and of Ax and Ay. What do you deduce?
- Compute the lengths of
x and of Ax. What characteristic of the linear transformation
x --> Ax does this reveal?
- Find the eigenvalues of
A, and take the absolute value of each eigenvalue. What do you observe?
How is this observation related to the preceding step? (Hint: How is
the length of Ax related to the length of x and the magnitude of lambda?)
- The preceding step also
tells you something about the absolute value of the determinant of A. What
is it? What does that imply about possible values for the determinant?
Which of these values actually is the determinant of A?
- Re-enter the commands starting
from step 1 to repeat all the steps with different random vectors. Check
that your observations are correct in this case as well -- or modify them
as necessary to account for both cases.
modules at math.duke.edu
