 
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
