Eigenvectors and Eigenvalues

Finding eigenvectors and eigenvalues is hard. Our general strategy was:

Compute the characteristic polynomial. For an n x n matrix, this involves taking the determinant of an n x n matrix with entries polynomials, which is slow. (The fast method for computing determinants, row reduction, doesn’t help much since the entries are polynomials.)
Find the roots of the characteristic polynomial. You only know a closed form formula for the case n=2. There exist formulas in terms of roots for n=3 and 4, but not for n=5 or higher. You can always approximate the roots using Newton’s method, say, but it’s at best somewhat tedious.
For each root (eigenvalue), find the corresponding eigenvectors. This involves row reducing a matrix whose entries are perhaps complicated real numbers, once for each eigenvalue.

Fortunately, since finding eigenvalues and eigenvectors is important in practice, there are lots of techniques, and computer code for doing it, symbolically (precisely) and approximately, for general matrices and special classes.

Philosophy over; here are some commands to get started.

Before getting started, to make the output readable, type:
%typeset_mode True

Eigenvalues and eigenvectors in one step.

Let’s create the matrix from Example 5.1.4 in the text, and find its eigenvalues and eigenvectors it:
M = matrix([[4,-1,6],[2,1,6],[2,-1,8]]); M M.eigenvectors_right()

Here, CoCalc gives us a list of triples (eigenvalue, eigenvectors forming a basis for that eigenspace, algebraic multiplicity of the eigenspace). You’re probably most interested in the first two entries at the moment. (As usual, these are column vectors even though Sage displays them as rows.)

We can test these are eigenvectors, by multiplying M by them. For example:
M*vector([1,0,-1/3]) M*vector([1,0,-1/3])==2*vector([1,0,-1/3])

If you just want the list of eigenvalues, type:
M.eigenvalues()

In M.eigenvectors_right(), The word “right” is because we want vectors v so that Mv=cv, not vM=cv. (Why don’t we have to write M.eigenvalues_right()?)

The characteristic polynomial.
M.characteristic_polynomial()

To get the roots of the characteristic polynomial, you can type:
M.characteristic_polynomial().roots()

Sage gives you pairs (root, multiplicity of that root). So, in this case, 2 occurs as a root twice, while 9 occurs once. That is, if we factor the characteristic polynomial (M.characteristic_polynomial().factor() ) we get:

Warning! Sage seems to define the characteristic polynomial to be det(xI – A), while the book and WebWorks use det(A-xI).

Diagonalizing
Not every matrix is diagonalizable, but every matrix has a “Jordan normal form” (which we will not discuss, alas). If the M is diagonalizable then M’s Jordan normal form is diagonal (and conversely). So, to test if M is diagonalizable, you can use:
M.jordan_form()