Com S 633: Randomness in Computation Lecture 18 Scribe: Rakesh Setty Summary: Com S 633: Randomness in Computation Lecture 18 Scribe: Rakesh Setty 1 Spectral Expansion In the last class, we mentioned a theorem which states that for every n × n real symmetric matrix M, there exists orthogonal vectors v1 , v2 , . . . vn that are eigen vectors of the matrix M. Let 1 , 2 , . . . n be the corresponding eigen values. Recall that these values may not be distinct. From now, we use the following convention: We assume that all eigen vectors are ordered according to their absolute values. That is we have |1| |2| · · · |n|. For every v Rn v = a1v1 + a2v2 + . . . anvn Now, Mv = a1Mv1 + a2Mv2 + . . . anMvn = a11v1 + a22v2 + . . . annvn = 1a1v1 + 2a2v2 + . . . nanvn That is if we represent v with respect to eigen basis of M, then M is just stretching the vector v along each co-ordinate. Observation: v, ||Mv||2 |1| ||v||2 Now, look at the eigen space of 1. Consider a vector vP that is perpendicular to this sub space. Thus vP can be expressed as Collections: Computer Technologies and Information Sciences