- Learn to find complex eigenvalues and eigenvectors of a matrix.
In Section 5.4, we saw that a matrix whose characteristic polynomial has distinct real roots is diagonalizable: it is similar to a diagonal matrix, which is much simpler to analyze. In this section, we study matrices whose characteristic polynomial has complex roots. If a matrix has distinct complex eigenvalues, then it is also diagonalizable, but it similar to a diagonal matrix with complex entries. The geometric interpretation of such a matrix is a subtle question, which is treated in detail in the full version of the book.
See Appendix A for a review of the complex numbers.
As a consequence of the fundamental theorem of algebra as applied to the characteristic polynomial, we see that:
Every matrix has exactly complex eigenvalues, counted with multiplicity.
We can compute a corresponding (complex) eigenvector in exactly the same way as before: by row reducing the matrix Now, however, we have to do arithmetic with complex numbers.
If is a matrix with real entries, then its characteristic polynomial has real coefficients, so this note implies that its complex eigenvalues come in conjugate pairs. In the first example, we notice that
In the second example,
In these cases, an eigenvector for the conjugate eigenvalue is simply the conjugate eigenvector (the eigenvector obtained by conjugating each entry of the first eigenvector). This is always true. Indeed, if then
which exactly says that is an eigenvector of with eigenvalue
Let be a matrix with real entries. If
In other words, both eigenvalues and eigenvectors come in conjugate pairs.
Since it can be tedious to divide by complex numbers while row reducing, it is useful to learn the following trick, which works equally well for matrices with real entries.
Let be a matrix, and let be a (real or complex) eigenvalue. Then
assuming the first row of is nonzero.
Indeed, since is an eigenvalue, we know that is not an invertible matrix. It follows that the rows are collinear (otherwise the determinant is nonzero), so that the second row is automatically a (complex) multiple of the first:
It is obvious that is in the null space of this matrix, as is for that matter. Note that we never had to compute the second row of let alone row reduce!
In this example we found the eigenvectors and for the eigenvalues and respectively, but in this example we found the eigenvectors and for the same eigenvalues of the same matrix. These vectors do not look like multiples of each other at first—but since we now have complex numbers at our disposal, we can see that they actually are multiples: