#MATRIX ALGEBRA: DEFINITIONS AND BASIC FUNCTIONS
#SCALAR:
a=5;b=8;c=3.14
a
b
c

#VECTOR:
x1=c(3,-1,4)
x1
x2=c(1,2,-2)
x2

#MATRIX:
A=matrix(c(0,3,1,1,-1,1),nrow=2,ncol=3,byrow=T)
A

B=matrix(c(1,2,-2,5,-3,1),nrow=2,ncol=3,byrow=F,
 dimnames=list(c("row1","row2"),c("C.1","C.2","C.3")))
B

#VECTOR ADDITION AND SCALAR MULTIPLICATION:
x1+x2
c*x2

#VECTOR YRANSPOSE & MULTIPLICATION:
t(x1)        #transpose
x1*x2        #product by row
t(x1)%*%x2   #dot product

x1%*%t(x2)   #cross product
outer(x1,x2) #Equivalent to line above

#VECTOR LENGTH:
length(x1)        #number of elements in a vector
sqrt(t(x1)%*%x1)  #Euclidean length 

#ANGLE BETWEEN VECTORS:
cos.theta=(t(x1)%*%x2)/((sqrt(t(x1)%*%x1))*(sqrt(t(x2)%*%x2)))
cos.theta
theta.rad=acos(cos.theta)
theta.rad
theta=(180/pi)*theta.rad
theta
pi

#PROJECTION OF VECTORS:
proj12=((t(x1)%*%x2)/(t(x2)%*%x2))*x2
proj12

proj21=((t(x2)%*%x1)/(t(x1)%*%x1))*x1
proj21

#HANDY PROJECTION FUNCTION:
#Projects x onto y
proj <- function(x,y){
res=((t(x)%*%y)/(t(y)%*%y))*y
return(res)
}

proj(x1,x2)
proj(x2,x1)

#MATRIX SUMMARY INFORMATION:
nrow(A)
ncol(B)
qr(A)$rank

#MATRIX MULTIPLICATION:
A%*%B
t(A)%*%t(B)
A%*%t(B)
t(A)%*%B
t(B)%*%A
t(x1)%*%A
A%*%x1
t(x2)%*%t(B)

#IDENTITY MATRIX:
I4=diag(4)
I4

#MATRIX TRACE:
is.square <- function(M) {
return (is.matrix(M) && (nrow(M) == ncol(M)));
}
trace <- function(M) {
return (ifelse(is.square(M),sum(diag(M)),NA));
}
trace(S)

#MATRIX INVERSE:
S=matrix(c(4,7,6,5,1,1,3,2,1),nrow=3,ncol=3,byrow=T)
Sinv=solve(S)
Sinv

#QUADRATIC FORM:
M=matrix(c(1,-5,7,-5,2,4,7,4,3),nrow=3,ncol=3,byrow=T)
t(x1)%*%M%*%x1