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Positive Definite Matrices and Minima

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Session Overview

Figure excerpted from 'Introduction to Linear Algebra' by G.S. Strang

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.

Session Activities

Lecture Video and Summary

Suggested Reading

  • Read Section 6.5 in the textbook.

Problem Solving Video

  • Watch the recitation video on Positive Definite Matrices and Minima (00:12:49)
  • Recitation video transcript (PDF)

Check Yourself

Problems and Solutions

Work the problems on your own and check your answers when you're done.

 

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