Advanced Linear Algebra - Advanced Linear Algebra, Lecture 6.1: Quadratic forms

Advanced Linear Algebra, Lecture 6.1: Quadratic forms A linear map H is self-adjoint, or Hermitian, if M*=M, and anti-self-adjoint, or anti-Hermitian, if M*=-M. Every linear map M can be decomposed into a sum M=H+A of a self-adjoint and anti-self-adjoint map, where H=(M+M*)/2 and A=(M-M*)/2. As we will see in the next lecture, self-adjoint maps have the special property that all of their eigenvalues are real, and the underlying vector space has an orthonormal basis of eigenvectors. A quadratic form is a function of the form q(x)=(x,Hx), where H is a self-adjoint linear map. Multiplying this out gives n^2 terms, but if we diagonalize H, then it only involves n terms. We will motivate quadratic forms by seeing how arise in the 2nd order terms of multivariate Taylor series. Finally, we will conclude with an example of a level set of a quadratic form, which is the equation of an ellipse. This does not become apparent until we diagonalize it, and graph it with respect to the eigenvector basis. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html