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Section5.5Complex Eigenvalues

Objectives
  1. 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.

Subsection5.5.1Matrices with Complex Eigenvalues

As a consequence of the fundamental theorem of algebra as applied to the characteristic polynomial, we see that:

Every n × n matrix has exactly n complex eigenvalues, counted with multiplicity.

We can compute a corresponding (complex) eigenvector in exactly the same way as before: by row reducing the matrix A − λ I n . Now, however, we have to do arithmetic with complex numbers.

Example(A 2 × 2 matrix)
Example(A 3 × 3 matrix)

If A 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

1 + i hasaneigenvector v 1 = E i 1 F 1 − i hasaneigenvector v 2 = E − i 1 F .

In the second example,

4 + 3 i 5hasaneigenvector v 1 = C − 12 − 9 i − 9 + 12 i 25 D 4 − 3 i 5hasaneigenvector v 2 = C − 12 + 9 i − 9 − 12 i 25 D

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 Av = λ v then

A v = Av = λ v = λ v ,

which exactly says that v is an eigenvector of A with eigenvalue λ .

Let A be a matrix with real entries. If

λ isacomplexeigenvaluewitheigenvector v ,then λ isacomplexeigenvaluewitheigenvector v .

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.

Eigenvector Trick for 2 × 2 Matrices

Let A be a 2 × 2 matrix, and let λ be a (real or complex) eigenvalue. Then

A − λ I 2 = E zw AA F = ⇒ E − wz F isaneigenvectorwitheigenvalue λ ,

assuming the first row of A − λ I 2 is nonzero.

Indeed, since λ is an eigenvalue, we know that A − λ I 2 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:

E zw AA F = E zwczcw F .

It is obvious that A − wz B is in the null space of this matrix, as is A w − z B , for that matter. Note that we never had to compute the second row of A − λ I 2 , let alone row reduce!

Example(A 2 × 2 matrix, the easy way)

In this example we found the eigenvectors A i 1 B and A − i 1 B for the eigenvalues 1 + i and 1 − i , respectively, but in this example we found the eigenvectors A 1 − i B and A 1 i B 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:

− i E i 1 F = E 1 − i F i E − i 1 F = E 1 i F .