Linear Algebra 7.4 Applications of Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors aren’t just abstract linear algebra tools—they’re the engine behind some of the most important models in science and engineering. In this lesson, we explore how these ideas help us understand long‑term population behavior, solve systems of linear differential equations, and rotate conic sections to reveal their true geometric form.

We begin with population growth models to show how eigenvalues predict long‑term growth or decline and how dominant eigenvectors reveal stable population distributions. Then we move to systems of linear differential equations, where eigenvalues determine the type of solution (exponential growth/decay, oscillations, spirals), and eigenvectors give the directions of motion in the phase plane. Finally, we connect eigenvectors to geometry by rotating conic sections, using diagonalization to eliminate cross‑terms and classify ellipses, hyperbolas, and parabolas.

Video Chapters:
00:00 Intro
00:07 Population Growth
2:57 Stable Age Distribution Vector
3:52 Solve a System of Linear Differential Equations
6:50 Rotation of a Conic
14:52 Optional – How we previously rotated the same conic

Textbook: Elementary Linear Algebra 8e, Ron Larson

Slide Decks: https://bellevueuniversity-my.sharepoint.com/:f:/g/personal/kbrehm_bellevue_edu/IgDVnTt2BohsRLsMfseN5FijAXYDd49YlIkHeq5cfmCxG2g?e=ZwTt3c

Playlist: https://youtube.com/playlist?list=PLl-gb0E4MII1RjTV2GegmXFTJwTrek8OI&si=g3Xc3yHB5GkmxMF4