14.2 Eigenvectors of nonsymmetric matrices

To find the eigenvectors of a nonsymmetric matrix,

1.

Find the eigenvalues of the matrix as described.

2.

For each distinct eigenvalue $\lambda_i$, find the basis of the null space of $A-\lambda_iI$. This basis forms a complete set of eigenvectors $\vec e_i$ for the eigenvalue. The method for finding nullspaces was discussed earlier.

There is always at least one eigenvector. However, for a multiple eigenvalue, the number of eigenvectors might be less than the multiplicity of the eigenvalue. If so, the matrix is defective. (This is not to be confused with singular. A matrix is singular if an eigenvalue is zero, making its determinant zero.) For example, a matrix is defective if a double eigenvalue has only one eigenvector. Or a triple eigenvector only one or two.