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Appendix B and 5.5 Complex Eigenvalues

HW 26 + HW 27 due together

handwritten only, see course page (do # 1, 2, 6, 8)

Introduction to Complex Numbers

A complex number is written as \( z = a + bi \), where \( a, b \) are real numbers.

\( b \) is the Imaginary part of \( z \): \( \text{Im}(z) \)

\( a \) is the real part of \( z \): \( \text{Re}(z) \)

\[ i^2 = -1 \]

Sets and Vector Spaces

  • \( \mathbb{R} \) : set of all real numbers
  • \( \mathbb{C} \) : set of all complex numbers
\[ \mathbb{R}^2 = \begin{bmatrix} x \\ y \end{bmatrix} \quad x, y \text{ real} \]
\[ \mathbb{C}^2 = \begin{bmatrix} z_1 \\ z_2 \end{bmatrix} \quad z_1, z_2 \text{ are complex} \]

Complex Conjugate

\( z = a + bi \)

\( \overline{z} = a - bi \)

\( \overline{3 + 4i} = 3 - 4i \)

\( \overline{3 - 4i} = 3 + 4i \)

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Arithmetic Operations

Addition and Subtraction

\[ (a + bi) \pm (c + di) = (a + c) \pm (b + d)i \]

Multiplication

\[ \begin{aligned} (a + bi)(c + di) &= ac + adi + cbi + bdi^2 \\ &= (ac - bd) + (ad + cb)i \end{aligned} \]

Division

Division is not like polynomials

\[ \begin{aligned} \frac{1 + 2i}{3 + 4i} &= \frac{1 + 2i}{3 + 4i} \cdot \frac{3 - 4i}{3 - 4i} = \frac{(1 + 2i)(3 - 4i)}{(3 + 4i)(3 - 4i)} \\ &= \frac{3 - 4i + 6i - 8i^2}{9 - 12i + 12i - 16i^2} = \frac{11 + 2i}{25} = \frac{11}{25} + \frac{2}{25}i \end{aligned} \]

Geometric Interpretation

\( a + bi \) can be interpreted like a vector in \( \mathbb{R}^2 \).

\( z = a + bi \)

\( \phi = \tan^{-1}\left(\frac{b}{a}\right) \)

modulus of \( z \) : \( |z| = \sqrt{a^2 + b^2} \)

Vector z = a + bi in the complex plane with axes \text{Re}(z) and \text{Im}(z) , showing angle \phi and coordinates a, b .
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Complex Numbers and Eigenvalues

Polar form: z = a + bi = r e^{i \phi} where r, \phi defined as on last page.

\[ z = r (\cos \phi + i \sin \phi) \]

de Moivre's Theorem

If \( z = r (\cos \phi + i \sin \phi) \), then

\[ z^k = r^k (\cos k\phi + i \sin k\phi) \]

5.5 Complex Eigenvalues

Consider the matrix:

\[ A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]

What does this do to \( \vec{x} \)?

\[ A \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]\[ A \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \end{bmatrix} \]
A 2D coordinate system showing a 90-degree counterclockwise rotation of basis vectors.
Figure 1: Geometric interpretation of matrix A as a rotation.

This is a quarter circle turn counterclockwise.

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\[ A \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -y \\ x \end{bmatrix} \]

No vector can preserve heading.

\[ A\vec{x} = \lambda\vec{x} \quad \lambda = ? \]

Find \( \lambda \)'s:

\[ |A - \lambda I| = 0 \quad A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]\[ \begin{vmatrix} -\lambda & -1 \\ 1 & -\lambda \end{vmatrix} = \lambda^2 + 1 = 0 \rightarrow \lambda = \pm i \]

Find Eigenvectors

\( \lambda = i \)

\( (A - \lambda I)\vec{x} = \vec{0} \)

\[ \begin{bmatrix} -i & -1 & 0 \\ 1 & -i & 0 \end{bmatrix} \sim \begin{bmatrix} 1 & -i & 0 \\ 0 & 0 & 0 \end{bmatrix} \]

\( x_2 \) free, \( x_1 = i x_2 \). Choose \( x_2 = 1 \).

\[ \vec{v} = \begin{bmatrix} i \\ 1 \end{bmatrix} \]

\( \lambda = -i \)

\( (A - \lambda I)\vec{x} = \vec{0} \)

\[ \begin{bmatrix} i & -1 & 0 \\ 1 & i & 0 \end{bmatrix} \sim \begin{bmatrix} 1 & i & 0 \\ 0 & 0 & 0 \end{bmatrix} \]

\( x_2 \) free, \( x_1 = -i x_2 \). Choose \( x_2 = 1 \).

\[ \vec{v} = \begin{bmatrix} -i \\ 1 \end{bmatrix} \]
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Complex Eigenvalues and Rotation

Note: \(\lambda\)'s and \(\vec{v}\)'s are complex conjugate pairs.

Complex \(\lambda\)'s are associated with rotation, but there can be scaling, too.

\[ A = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} \quad A \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \quad A \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \end{bmatrix} \]
A 2D coordinate system showing vectors [1,0], [0,1], and their transformations [1,1] and [-1,1].

45° ccw turn with a lengthening by a factor of \(\sqrt{2}\)

Eigenvalues tell us this

\[ \begin{vmatrix} 1-\lambda & -1 \\ 1 & 1-\lambda \end{vmatrix} = 0 \implies (1-\lambda)^2 + 1 = 0 \]
\[ 1-\lambda = i \quad \text{or} \quad 1-\lambda = -i \]\[ \lambda = 1-i \quad \text{or} \quad \lambda = 1+i \]

If \(\lambda = a \pm bi\), then the scaling factor is \(\sqrt{a^2 + b^2}\).

\(\phi = \tan^{-1}(\frac{b}{a})\) is the turning angle.

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Rotation Matrix Form

Rotation matrix: \(\begin{bmatrix} a & -b \\ b & a \end{bmatrix}\), where \(a, b\) are real and not both zero.

\[ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} = r \begin{bmatrix} a/r & -b/r \\ b/r & a/r \end{bmatrix} \]\[ = \begin{bmatrix} r & 0 \\ 0 & r \end{bmatrix} \begin{bmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{bmatrix} \]
A point (a,b) in the Cartesian plane with radius r and angle phi.

\(r = \sqrt{a^2 + b^2}\) is the scaling factor.

\(\phi = \tan^{-1}(\frac{b}{a})\) is the turning angle.

Example

\[ A = \begin{bmatrix} 1 & -2 \\ 1 & 3 \end{bmatrix} \quad \begin{matrix} \lambda = 2+i \\ \lambda = 2-i \end{matrix} \quad \begin{matrix} \vec{v} = \begin{bmatrix} -1+i \\ 1 \end{bmatrix} \\ \vec{v} = \begin{bmatrix} -1-i \\ 1 \end{bmatrix} \end{matrix} \]

\(\lambda = a \pm bi\). Here, \(a=2, b=1\). Scaling factor \(r = \sqrt{a^2 + b^2} = \sqrt{5}\).

Define \(C = \begin{bmatrix} a & -b \\ b & a \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 1 & 2 \end{bmatrix}\). This is hidden inside \(A\).

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Matrix Decomposition and Rotation

We can decompose matrix \(A\) much like how we diagonalize matrices:

\[A = P C P^{-1}\]

Note: \(C\) is not diagonal, but a rotation matrix.

Eigenvectors and Real/Imaginary Parts

Given eigenvectors:

\[\text{eigenvectors: } \begin{bmatrix} -1 + i \\ 1 \end{bmatrix}, \begin{bmatrix} -1 - i \\ 1 \end{bmatrix}\]

Separate real and imaginary parts:

\[\vec{v} = \begin{bmatrix} -1 \\ 1 \end{bmatrix} \pm i \begin{bmatrix} 1 \\ 0 \end{bmatrix}\]

Use these as columns of \(P\).

Constructing the Matrices

\[P = \begin{bmatrix} -1 & 1 \\ 1 & 0 \end{bmatrix} \quad P^{-1} = \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix} \quad C = \begin{bmatrix} 2 & -1 \\ 1 & 2 \end{bmatrix}\]

Final decomposition:

\[A = \begin{bmatrix} 1 & -2 \\ 1 & 3 \end{bmatrix} = P C P^{-1} = \begin{bmatrix} -1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}\]

The product \(C P^{-1}\) represents scaling and rotation.

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Geometric Interpretation of Matrix Transformation

\[A\vec{x} = P C P^{-1} \vec{x}\]
  • \(P^{-1} \vec{x}\): Change of coordinates/variables
  • \(C (P^{-1} \vec{x})\): Scale, rotate
  • \(P (C P^{-1} \vec{x})\): Undo change of variables/coordinates

Complex Eigenvalue Pairs

\(\lambda\), if complex, must come in pairs.

\[A = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 5 & -3 \end{bmatrix}\]

Eigenvalues (\(\lambda\))

\(\lambda = \underbrace{-1 + i, -1 - i}_{\text{pairs of complex } \lambda\text{'s}}, 4\)

Eigenvectors (\(\vec{v}\))

\[\vec{v} = \underbrace{\begin{bmatrix} 0 \\ 2/5 + 1/5i \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 2/5 - 1/5i \\ 1 \end{bmatrix}}_{\text{pairs}}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\]