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 v 1 , v 2 , . . . v n 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 # R n v = a 1 v 1 + a 2 v 2 + . . . a n v n Now, Mv = a 1 Mv 1 + a 2 Mv 2 + . . . a n Mv n = a 1 # 1 v 1 + a 2 # 2 v 2 + . . . a n # n v n = # 1 a 1 v 1 + # 2 a 2 v 2 + . . . # n a n v n That is if we represent v with respect to eigen basis of M , then M is just stretching the vector v along each co­ordinate. Observation: #v, ||Mv|| 2 # |# 1 | ||v|| 2 Now, look at the eigen space of # 1 . Consider a vector v P that is perpendicular to this sub space. Thus v P can be expressed as Collections: Computer Technologies and Information Sciences