 
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 coordinate.
Observation: v, Mv2 1 v2
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
