Press BACK to return to the lab.

Solution to Eigenvectors

The matrix given is:

From the previous section, you should have found that the eigenvalues for this matrix are = -1 or 8.

Using Octave,

octave:1> B = [3 2 4; 2 0 2;4 2 3]
octave:2> [x, lambda] = eig(B)
x =

0.74536 -0.66667 -0.34487
-0.29814 -0.33333 -0.65498
-0.59628 -0.66667 0.67236

lambda =

-1.00000 0.00000 0.00000
0.00000 8.00000 0.00000
0.00000 0.00000 -1.00000

So the answer from Octave gives two unit eigenvectors for = -1, and one solution for = 8.

Calculation by hand, with = -1, will be:

Reading from the matrix,

Remember that = -1 appears as two of the three eigenvalues, so there are two degrees of freedom for the solution, i.e. the solution is actually a plane defined by this equation.

So any vector on this plane is an eigenvector of B:

The answers, (0.74536, -0.29814, -0.59628) and (-0.34487, -0.65498, 0.67236), shown in Octave for = -1 are only two of the possible unit eigenvectors for the solution. They should satisfy the equation of the plane.

For = 8, the same method can be used to find the eigenvector:

Reading from the matrix:

Letting = 2, then = 1 and = 2. This answer is a multiple of the solution given by Octave, which is (-0.66667, -0.33333, -0.66667). Note that = 8 has only one degree of freedom.