28.1 Case Studies of Eigenvalues of 2x2 Matrices Analysis, Categories, and Relations

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: 6.7 Linearly Independent Vectors
Questions: N/A

In our introduction to eigenvalue theory, we will build intuition using the processes of inductive reasoning, where we learn through example before studying the theory. We begin this work by conducting cases studies for the eigenvalue and eigenvector information of a collection of $2 \times 2$ matrices.

Matrix Analysis

Matrix Structure

What type(s) of structure does the matrix have?
- 8.5 Diagonal Matrix
- 8.9 Upper-Triangular Matrix
- 8.7 Lower-Triangular Matrix
- Symmetric
- Positive Definite (or positive semidefinite)
- Invertible

Algebraic properties of eigenvalues and eigenvectors

What are the eigenvalues and eigenvectors?
How many eigenvalues are there?
For each eigenvalue, how many eigenvectors can we find?
What kind of scalar is each eigenvalue?
- Real number (or complex number)
- Positive (or negative)
- Nonzero (or zero)
- Are any eigenvalues repeated?

Geometric Properties of eigenvalues and eigenvectors

What types of geometric intuition can we build based on our eigenvalue and eigenvector information?
When we think about the algebraic property that

A \vec{x} = λ \vec{x}

which of the following arises
- scaling (stretch or compress) along eigenvector
- reflection (flip orientation)
- rotation between input and output
- projection (loss of dimensions)

By the end of our case studies, we will have explored a large collection of $2 \times 2$ and $3 \times 3$ matrices, answering questions from each of our categories:

Matrix Structure
Algebraic Properties of eigen-information
Geometric Properties of eigen-information

Four Categories of

2 \times 2

Matrices

These are the four categories of $2 \times 2$ matrices based on the properties of the eigenvalues and eigenvectors. If $A \in R^{2 \times 2}$ is a "given" matrix and we find the eigenvalues and eigenvectors of $A$ , then there are four categories into which the properties of the eigen-information fit:

Matrix $A$ has $2$ distinct real eigenvalues and $2$ linearly independent eigenvectors.

λ_{1} \neq λ_{2} \in R and {\vec{v}}_{1}, {\vec{v}}_{2} \in R^{2} such that A {\vec{v}}_{k} = λ_{k} {\vec{v}}_{k}

Matrix $A$ has $1$ repeated real eigenvalue and $2$ linearly independent eigenvectors.

λ = λ_{1} = λ_{2} \in R and {\vec{v}}_{1}, {\vec{v}}_{2} \in R^{2} such that A {\vec{v}}_{k} = λ_{k} {\vec{v}}_{k}

Matrix $A$ has $1$ repeated real eigenvalue and only $1$ linearly independent eigenvalue.

λ = λ_{1} = λ_{2} \in R and {\vec{v}}_{1} \in R^{2} such that A {\vec{v}}_{1} = λ {\vec{v}}_{1}

Matrix $A$ has no real eigenvalues (but instead $2$ complex conjugate eigenvalues and $2$ linearly independent complex conjugate eigenvectors).

\begin{aligned} λ_{1} & = λ = a + b i \in C \\ λ_{2} & = λ = a + b i \in C \end{aligned}

where $i = \sqrt{- 1}$ and $a, b \in R$ and

\begin{aligned} {\vec{v}}_{1} & = v = \vec{x} + i \vec{y} \in C^{2} \\ {\vec{v}}_{2} & = v = \vec{x} - i \vec{y} \in C^{2} \end{aligned}

where $\vec{x}, \vec{y} \in R^{2}$ .