5.5 Complex Eigenvalues

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Section 5.5 Complex Eigenvalues

Handout 5.5 Complex Eigenvalues

Objectives: Topics

Complex numbers: addition, multiplication, complex conjugate

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Complex eigenvalues and eigenvectors

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Eigenvalue theorems

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Objectives: Goals

Use eigenvalues to identify the rotation and dilation of a linear transform.

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Recognize rotation–dilation matrices.

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Find complex eigenvalues and eigenvectors of a real matrix.

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Apply theorems to characterize matrices with complex eigenvalues.

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Example 5.41.

Consider the linear transformation \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) which rotates vectors counterclockwise by 90\(^\circ\text{.}\)

(a)

Find the matrix \(A\) so that \(T(\vec x) = A\vec x\text{.}\)

(b)

Find the characteristic polynomial of \(A\text{.}\)

(c)

What are the eigenvalues of \(A\text{?}\)

Definition 5.42. Complex Numbes.

The complex numbers are denoted by \(\mathbb{C}\text{,}\) where

\begin{equation*} \mathbb{C}=\{\,a+bi \mid a,b\in\mathbb{R}\,\}. \end{equation*}

We can identify \(\mathbb{C}\) with \(\mathbb{R}^2\) via \(a+bi \leftrightarrow (a,b)\text{.}\)

We can add and multiply complex numbers by treating \(i\) as variable and simplifying powers of \(i\) appropriately.

Example 5.43.

Let \(z_1 = 2 - 3i\) and let \(z_2 = -1 + i\text{.}\) Calculate the following:

(a)

Sum of complex numbers: \(z_1 + z_2\)

(b)

Product of complex numbers: \(z_1z_2\)

Definition 5.44. Complex Conjugate.

The conjugate of \(a+bi\) is \(\overline{a+bi}=a-bi\text{.}\)

Definition 5.45. Absolute Value (Modulus).

The absolute value (modulus) of \(a+bi\) is \(\lvert a+bi\rvert=\sqrt{a^2+b^2}\text{.}\)

Definition 5.46.

The polar form of a complex number is \(a+bi = r(\cos\phi + i\sin\phi) = re^{i\phi}\text{,}\) where \(r=\lvert a+bi\rvert\) and \(\phi\) is the argument.

Example 5.47.

Given \(z=3-4i\text{,}\) compute \(\overline{z}\text{,}\) \(\lvert z\rvert\text{,}\) and the polar form of \(z\text{.}\)

Theorem 5.48. Conjugate Properties.

Let \(x,y\in\mathbb{C}\) and \(\vec v\in\mathbb{C}^n\text{.,}\) Then we have the following properties:

\(\displaystyle \overline{(x+y)}=\overline{x}+\overline{y}\)

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\(\displaystyle \overline{(xy)}=\overline{x} \ \overline{y}\)

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\(\overline{A\vec v}=A\,\overline{\vec v}\) (for a real matrix \(A\))

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\(\displaystyle \operatorname{Im}(x\overline{x})=0\)

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Conjugation reflects points across the real axis.

$Diagram showing complex conjugation as reflection over real axis.$

The point \(z=x+iy\) at angle \(\phi\) reflects to \(\overline{z}=x-iy\) at angle \(-\phi\) across the real axis.

Suppose \(z_1\) has angle \(\phi_1\) and \(z_2\) has angle \(\phi_2\text{;}\) that is, let \(z_1 = |z_1|e^{i\phi_1}\) and \(z_2 = |z_2|e^{i\phi_2}\text{.}\) Note that

\begin{equation*} z_3=z_1z_2=(\lvert z_1\rvert e^{i\phi_1})(\lvert z_2\rvert e^{i\phi_2}) = \lvert z_1\rvert \lvert z_2\rvert e^{i(\phi_1+\phi_2)}. \end{equation*}

Therefore, the product \(z_1z_2\) has modulus \(|z_1||z_2|\) and has angle \(\phi_1 + \phi_2\text{.}\)

$Angles in polar form for complex numbers and their product.$

Vectors representing \(z_1\text{,}\) \(z_2\text{,}\) and their product \(z_3=z_1z_2\) with angles adding: \(\phi_3=\phi_1+\phi_2\text{.}\)

Theorem 5.49. Fundamental Theorem of Algebra.

Every polynomial of degree \(n\) has exactly \(n\) complex roots, counting multiplicity.

Theorem 5.50. Conjugate Pairs for Real Coefficients.

If \(\lambda\in\mathbb{C}\) is a root of a real polynomial \(p(x)\text{,}\) then the conjugate \(\overline{\lambda}\) is also a root of \(p(x)\text{.}\)

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If \(\lambda\) is an eigenvalue of a real matrix \(A\) with eigenvector \(\vec v\text{,}\) then \(\overline{\lambda}\) is an eigenvalue of \(A\) with eigenvector \(\overline{\vec v}\text{.}\)

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Example 5.51.

Four eigenvalues of a \(7\times 7\) real matrix \(A\) are \(-2\text{,}\) \(4+i\text{,}\) \(-4-i\text{,}\) and \(i\text{.}\) What are the other eigenvalues?

Example 5.52.

The matrix that rotates vectors by \(\phi=\pi/4\) about the origin and then scales by \(r=\sqrt{2}\) is

\begin{equation*} A= \begin{bmatrix} r \amp 0\\ 0 \amp r \end{bmatrix} \begin{bmatrix} \cos\phi \amp -\sin\phi\\ \sin\phi \amp \cos\phi \end{bmatrix} = \begin{bmatrix} 1 \amp -1\\ 1 \amp 1 \end{bmatrix}. \end{equation*}

Find the eigenvalues of \(A\) and one eigenvector for each eigenvalue.

The previous example is a special case of the real rotation–dilation matrix \(C=\begin{bmatrix} a \amp -b\\ b \amp a \end{bmatrix}\text{.}\)

Example 5.53.

Compute the eigenvalues of

\begin{equation*} C=\begin{bmatrix} a \amp -b\\ b \amp a \end{bmatrix} \end{equation*}

and express them in the polar form \(re^{i\phi}\text{.}\)

Example 5.54.

Find the complex eigenvalues and an associated complex eigenvector for each eigenvalue of

\begin{equation*} A=\begin{bmatrix} 1 \amp -2\\ 1 \amp 3 \end{bmatrix}. \end{equation*}