Part 1: Calculation of
Eigenvalues and Eigenvectors
The matrices A,
B, C, and the 3x3 identity matrix I3are
defined in your worksheet. Calculate the characteristic polynomial of A
by taking the determinant of A - lambda I3Factor
the polynomial.
Solve the linear system
(A - I3) v = 0 by finding the reduced row echelon form
of A - I3. Explain why this enables you to write down
an eigenvector corresponding to the eigenvalue 1.
Determine an eigenvector
v corresponding to the eigenvalue 1. Check that this is correct
by calculating Av and v. Describe all eigenvectors
of A that correspond to the eigenvalue 1.
Repeat the process in Steps
2 and 3 for the other eigenvalues of A, and record the results.
Now find the eigenvalues
of B, and for each eigenvalue describe all the corresponding eigenvectors.
Repeat this process for C. How do the calculations for A, B,
and C differ?
Now check your eigenvalue
and eigenvector calculations using the appropriate commands in your computer
algebra system. Explain any differences in the output of the commands for
matrices A, B, and C.