Eigenvalues
and Eigenvectors
Part 1: Commands related
to eigenvalues and eigenvectors
We start by exploring the
commands available in your favorite computer algebra system. Because these
commands vary from one system to another, some of the instructions will
be given in the prepared file and described here only in general terms.
- Enter the 3 x 3 matrix
A. This matrix is simple enough that you can calculate its characteristic
polynomial by hand. Do so.
- Now enter the commands
that will construct the characteristic matrix and its determinant, the
characteristic polynomial.
- Note: In some texts
this polynomial is defined as det(A - xI) and in some texts as det(xI -
A). The distinction is not important because a change of sign throughout
an n x n matrix changes its determinant by a factor of (-1)n,
and that does not change the roots of the polynomial at all. Whichever
form your computer algebra system uses, go with it.
- Finish calculating the
eigenvalues of A by hand: Factor your characteristic polynomial, and write
down its roots. Then use your computer algebra system to check your work.
- Now use the command in
your computer algebra system to calculate eigenvalues of A directly.
- One of the eigenvalues
of A is 6. Find a basis for the null space of 6I - A, and explain what
your result means in terms of "eigenstuff."
- Next, use a command to
compute eigenvectors of A directly. Explain what you see as output.
- Finally, use a command
to compute both the eigenvalues of A and a matrix P whose columns are the
eigenvectors. (If no such command is available, build P from the eigenvectors
in the preceding step.) Calculate P-1AP,
and explain the result.
modules at math.duke.edu
