Time for action - using Octave for advanced linear algebra

1. 1. It is easy to calculate the determinant of a 2 x 2 matrix, but for a 3 x 3 matrix, the calculation becomes tedious, not to mention larger size matrices. Octave has a function det that can do this for you:

octave:46> A=[2 1 -3; 4 -2 -2; -1 0.5 -0.5];
octave:47>det(A)
ans = 8

• Recall from linear algebra that the determinant is only defined for a square n x n matrix. Octave will issue an error message if you pass a non-square matrix input argument.

1. 2. Let us change A a bit:

octave:48> A=[2 1 -3; 4 -2 -2; -2 1 1];
octave:49>det(A)
ans = 0

• This result is consistent with the result from Chapter 2. A does not have full rank, that is, the determinant is 0.

1. 3. The eigenvalues of an n x n matrix are given by the equation:

(3.6)

• To calculate the eigenvalues in Octave, we can use the eig function:

octave:50> A = [1 2; 3 4];
octave:51>eig(A)
ans =
-0.3722
5.3722

1. 4. eig can also return the eigenvectors. However, given two...