A Revealing Introduction to Hidden Markov Models

http://www.cs.sjsu.edu/~stamp/RUA/HMM.pdf

Rank and Nullity

http://algebra.math.ust.hk/vector_space/10_rank/lecture1.shtml

Simple SAS codes to understand set loop and order duplicated cases.

data try;
input pid value;
datalines;
1 1
1 3
1 5
2 1
2 2
3 1
3 2
3 6
3 8
;
run;
data try2;
set try;
by pid;
episode+1;
if first.pid then episode=1;
run;

Spectral decomposition

A<-matrix(c(2,1,1,2),nrow=2,ncol=2)
A
#     [,1] [,2]
#[1,]    2    1
#[2,]    1    2
E<-eigen(A)
E
#$values
#[1] 3 1

#$vectors
#          [,1]       [,2]
#[1,] 0.7071068 -0.7071068
#[2,] 0.7071068  0.7071068


c1<-c(E$vectors[1,1],E$vectors[2,1])
c1
#[1] 0.7071068 0.7071068

c2<-c(E$vectors[1,2],E$vectors[2,2])
c2
#[1] -0.7071068  0.7071068

Q1<-c1%*%t(c1)
Q2<-c2%*%t(c2)

3*Q1+1*Q2
#     [,1] [,2]
#[1,]    2    1
#[2,]    1    2