# Symmetric matrices and Positive Definite matrices

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Lesson Summary:

In this lesson, we learn about the special properties of symmetric matrices and positive definite matrices. The main focus is on the eigenvalues and eigenvectors of symmetric matrices, which are all real and perpendicular, respectively. The eigenvectors can also be chosen to be unit vectors. This leads to a factorization of the matrix into an orthonormal matrix, diagonal matrix, and its transpose. The spectral theorem, which is a famous theorem of linear algebra, tells us that a symmetric matrix can be factored in this way. We also learn that good matrices are those that have real eigenvalues and perpendicular eigenvectors, which are symmetric if real.

Lesson Description:

Symmetric matrices and Positive Definite matrices -- Lecture 25. Special properties of these matrices and why they are important.

Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 23, 2008). License: Creative Commons BY-NC-SA.