Advanced Linear Algebra - Advanced Linear Algebra, Lecture 6.4: The Rayleigh quotient

Advanced Linear Algebra, Lecture 6.4: The Rayleigh quotient The Rayleigh quotient of a self-adjoint map H is the function R(x)=(x,Hx)/(x,x), defined on non-zero vectors. It is clear that R(x)=R(kx), and so we only need to consider it on the unit sphere. It is elementary to show that if Hv=λv, then R(v)=λ, and that the maximum and minimum value of R(x) are precisely the largest and smallest eigenvalues, respectively. In fact, more is true: the critical points of the Rayleigh quotient occur at the eigenvectors. This is useful in numerical linear algebra because it is a 2nd order Taylor approximation of the eigenvalues. Using the Rayleigh quotient, we can explicitly construct an orthogonal basis of eigenvectors, and get an alternative proof of the spectral resolution of self-adjoint maps, that doesn't rely on the fundamental theorem of algebra. We also derive a "min-max principal", which says that the k'th largest eigenvalue can be found by minimizing the maximum value of R(x) over all k-dimensional subspaces of X. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html