Linear Algebra 7.3 Symmetric Matrices and Orthogonal Diagonalization

Unlock one of the most elegant results in linear algebra: **every real symmetric matrix can be orthogonally diagonalized**. In this lesson, we break down *why* symmetric matrices behave so nicely, *how* orthogonal diagonalization works, and *what* it means geometrically.

Whether you’re in a first linear algebra course or brushing up for applications in data science, differential equations, or machine learning, this video gives you the intuition and the step‑by‑step structure you need.

Video Chapters:
00:00 Intro
00:08 Symmetric Matrices
02:36 Diagonalizing a Symmetric Matrix (3×3)
09:27 Finding Eigenvalues (4×4)
12:23 Orthogonal Matrices
13:03 An Orthogonal Matrix
13:52 An Orthonormal Set
16:05 Orthogonal Matrices (on your own)
18:56 Orthogonal Diagonalization (2×2)
22:25 Orthogonal Diagonalization (3×3)
35:38 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