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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.

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), 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 (-0.66667, -0.33333, -0.66667). Note that = 8 has only one degree of freedom.