Linear Algebra 7.2 Diagonalization

In this lesson, we break down one of the most elegant ideas in linear algebra: diagonalization. You’ll learn what it means for a matrix to be diagonalizable, why diagonalization matters, and how to actually compute a diagonalization step‑by‑step.

We walk through:
• What it means for a matrix to be similar to a diagonal matrix
• How eigenvalues and eigenvectors determine diagonalizability
• How to construct the matrices P and P^-1
• What to do when a matrix is not diagonalizable
• Fully worked examples to build confidence
By the end of the video, you’ll understand not just the procedure, but the why behind it—setting you up for success in later topics like matrix powers, differential equations, and linear transformations.

Video Chapters:
00:00 Intro
01:14 Diagonalization 2X2
07:03 Diagonalization 2X2 – Doesn’t work
8:56 Diagonalization 3X3 – Finding the eigenvalues
13:54 Diagonalization 3X3 – Finding the eigenvectors
17:46 Diagonalization 3X3 – Finishing up
18:47 Diagonalization 3X3 – Doesn’t work
20:18 Diagonalization Within A Linear Transformation
24:28 Up Next

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