Lecture 58. Signature of a bilinear form

0:00 Eigenvalues of a bilinear form
4:32 Definition of the signature of a bilinear form
6:14 Theorem on independence of the signature on a choice of a basis
7:53 Kernel of a bilinear form
23:52 Proof of the main theorem
33:55 Determinants of principal minors
35:20 Sylvester's Criterion
48:26 Examples of computation of the signature
51:49 Multivariable second derivative test

In this lecture we introduce the notion of a signature of a bilinear (or a quadratic) form. We show that unlike eigenvalues, the signature is independent of a choice of a basis. We also present Sylvester's Criterion, which is helpful for computing the signature. We conclude this lecture by showing how the signature of a quadratic form is used to analyse local maxima and minima of multivariable functions.
This is a lecture in a "Linear Algebra" course for students specializing in mathematics.