27.1 The Standard Eigenvalue Problem

Types: N/A
Examples: 27.2 Intro to Coupled Pendula Problem
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: N/A
Questions: N/A

The Standard Eigenvalue Problem

"Given" a square matrix $A \in R^{n \times n}$ , find all scalars $λ$ and associated $n \times 1$ vectors $\vec{x}$ such that

A \cdot \vec{x} = λ \vec{x}

$A$ is a square matrix arising from a modeling context
$λ$ is a scalar eigenvalue
$\vec{x}$ is the eigenvector associated with eigenvalue $λ$

Remark. The system only scales the vector $\vec{x}$ , it does not fundamentally change its values. A nice analogy is to think of $\vec{x}$ as your voice into a karaoke machine, and $λ$ being the volume you want the system to output.
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Remark. You may be wondering, "Why should I care about Eigenvalues?" Below are some examples of fields that use Eigenvalues.

Vibrations Analysis: the study of things that shake or move back and forth... the study of objects that vibrate
1. Suspension system in vehicles
2. Brake system in vehicles
3. Motion of buildings in earthquakes
4. Motion of bridges in wind
5. Motion of wings of plane in flight
Electric Circuit Analysis and Design
Image Compression Algorithms
Machine Learning & Data Analysis (Principal Component Analysis)
Google's Page Rank Algorithm
Netflix's old move rating predictions
The foundation of theory for Finite Element Analysis

There are two common themes that arise across these applications:

~ Continuous Problems
Many of these problems exist in time and space. They involve math that is built on top of Calculus (equations with continuous functions) not algebra (equations with a collection of scalars)
~ Large number of dimensions
Many of these problems involve systems having huge number of components with a goal of reducing complexity/dimensions.

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