#R example 2, 1/24/06

################-------vectors and operations-----

# 1) ----generate vectors--------------------

c(1,-2,1)

x=c(1,-2,1)

y=1:3

y

z<-seq(-3,2,length=6)

z

seq(-3,2,by=1)

seq(length=6, from=-3, by=1)

rep(0,3)

rep(y,2)

# 2) ----operators--- +,-,*,/ and ^(power)------

2*x+y

2*x+y+1

x*y # component-wise product, still a vector

x%*%y # inner product = sum(x*y)

x/y

x^2 # x*x

sqrt(sum(x*x)) # norm of x

########-------------matrices and operations------------

# 1)----generate matrix--------------------

matrix(1:12, nrow=3)

matrix(1:12, ncol=4)

matrix(1:12,2)

matrix(1:12,2,byrow=T)

T12=read.table("D://T1-2.DAT")

T12 #data.frame

scan("D://T1-2.DAT")

matrix(scan("D://T1-2.DAT"), ncol=3)

T12m=matrix(scan("D://T1-2.DAT"), ncol=3, byrow=T)

T12m #matrix

# 2)----matrix operation------------------

T12+T12

T12m+T12m

T12*T12 #component-wise square

T12%*%T12 #not data matrix

T12m%*%T12m #not conformable

T4141=T12m%*%t(T12m)

T33=t(T12m)%*%T12m

t33=crossprod(T12m) #crossprod(A, B)=A'B

T33

t33

# 3)----inverse matrix-------------------

solve(T33)%*%T33

# 4)----eigenvalue and eigenvector--------

eigen(T33)

eigen(T33, only.values=T)

vv=eigen(T33)

vv$values

vv$vectors

# 5)----spetral decomposition (singular-value decomposition)

svd(T33) #svd(A)gives eigenvalues d, orthogonal matrices u and v such that A=u%*%diag(d)%*%t(v)

SD=svd(T33)

SD$u%*%diag(SD$d)%*%t(SD$v)

T33

#singular value decomposition can also handler no square matrix

x=c(1, 3, 4, 5, 0, -2)

x=matrix(x,ncol=2,byrow=T)

x

svd(x)