Let A be a Hermitian matrix with eigenvalues 1 2 n and orthonormal eigenvectors u1, ... , un. For any nonzero vector x in Rn, the Rayleigh quotient (x) is defined by (x) = Ax, x x, x = xHAx xHx (a) If x = c1u1 ++ cnun, show that (x) = |c1| 21 + |c2| 22 ++|cn| 2n c2 (b) Show that n (x) 1 (c) Show that max x=0 (x) = 1 and min x=0 (x) = n

Week 1 Monday, August 28, 2017 9:18 PM 1. Functions Situation where there is an input (x) and some output (y) represented by an ordered pair (x,y). A function is a relation. So, for every input there's exactly ONE output Ex. Function: Input a number, output its square. (1,1) (2,4) (3,9)… Ex. Not a function: In put a person, output their email (Jane Doe, Jdoe@student.college.edu) (Jane Doe, Jdoe@gmail.com) If an input x is related to output y by a function, call it f X is the independent variable Y is the dependent variable We will write y=f(x) Ex. Function that gives area A of a circle with radius r 2 A=∏r 1.1 Evaluat