In calculus, the second derivative decides whether a critical point of y(x) is a minimum. For functions of multiple variables, the test is whether a matrix of second derivatives is positive definite. In this session we learn several ways of testing for positive definiteness and also how the shape of the graph of ƒ(x) = xT Ax is determined by the entries of A.
Lecture Video and Summary
- Watch the video lecture Positive Definite Matrices and Minima (00:50:40)
Lecture 27: Positive Definite Matrices and Minima
- Read the accompanying lecture summary (PDF)
- Lecture video transcript (PDF)
- Read Section 6.5 in the 4th or 5th edition.
Problem Solving Video
- Watch the recitation video on Positive Definite Matrices and Minima (00:12:49)
Problem Solving: Positive Definite Matrices and Minima
- Recitation video transcript (PDF)
Problems and Solutions
Work the problems on your own and check your answers when you're done.