Lecture 58. Signature of a bilinear form скачать в хорошем качестве

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.