Lecture 5: Positive Definite and Semidefinite Matrices

Description

In this lecture, Professor Strang continues reviewing key matrices, such as positive definite and semidefinite matrices. This lecture concludes his review of the highlights of linear algebra.

Summary

All eigenvalues of S are positive.
Energy x^TSx is positive for x \(\neq 0\).
All pivots are positive S = A^TA with independent columns in A.
All leading determinants are positive 5 EQUIVALENT TESTS.
Second derivative matrix is positive definite at a minimum point.
Semidefinite allows zero evalues/energy/pivots/determinants.

Related section in textbook: I.7

Instructor: Prof. Gilbert Strang