HW2

#part1: 

# Answer1: 

A = matrix(c(1,4,2,7,0,-3,1,3,-2), nrow=3, byrow=TRUE) # matrix A
A

##      [,1] [,2] [,3]
## [1,]    1    4    2
## [2,]    7    0   -3
## [3,]    1    3   -2

At = t(A)  #transpose of matrix A
At

##      [,1] [,2] [,3]
## [1,]    1    7    1
## [2,]    4    0    3
## [3,]    2   -3   -2

A %*% At    # multiply matrix A by it transpose 

##      [,1] [,2] [,3]
## [1,]   21    1    9
## [2,]    1   58   13
## [3,]    9   13   14

At %*% A    # do the opposite multiply the matrix transpose by the matrix

##      [,1] [,2] [,3]
## [1,]   51    7  -21
## [2,]    7   25    2
## [3,]  -21    2   17

A %*% At == At %*% A # comparison , the results is false , which mean A^TA  not equal to AA^T

##       [,1]  [,2]  [,3]
## [1,] FALSE FALSE FALSE
## [2,] FALSE FALSE FALSE
## [3,] FALSE FALSE FALSE

#answer2: 

A= matrix(c(2,1,0,1,2,0,0,0,2), nrow=3, byrow=TRUE) # try symmetric matrix 
A

##      [,1] [,2] [,3]
## [1,]    2    1    0
## [2,]    1    2    0
## [3,]    0    0    2

At = t(A)  #transpose of matrix A
At

##      [,1] [,2] [,3]
## [1,]    2    1    0
## [2,]    1    2    0
## [3,]    0    0    2

A %*% At    # multiply matrix A by it transpose 

##      [,1] [,2] [,3]
## [1,]    5    4    0
## [2,]    4    5    0
## [3,]    0    0    4

At %*% A    # do the opposite multiply the matrix transpose by the matrix

##      [,1] [,2] [,3]
## [1,]    5    4    0
## [2,]    4    5    0
## [3,]    0    0    4

A %*% At == At %*% A # comparison , the results is true , when the matrix is symmetric A^TA is equal to AA^T

##      [,1] [,2] [,3]
## [1,] TRUE TRUE TRUE
## [2,] TRUE TRUE TRUE
## [3,] TRUE TRUE TRUE

#Part2:

factorizeLU <- function(A) {
  # see A is square
  if (dim(A)[1]!=dim(A)[2]) {
    return(NA)
  }
  
  U <- A
  n <- dim(A)[1]
  L <- diag(n)
  
  # If dimension is 1, then U=A and L=[1]
  if (n==1) {
    return(list(L,U))
  }
  
  # Determine the multiplier for each position and add it to L
  for(i in 2:n) {
    for(j in 1:(i-1)) {
      multiplier <- -U[i,j] / U[j,j]
      U[i, ] <- multiplier * U[j, ] + U[i, ]
      L[i,j] <- -multiplier
    }
  }
  return(list(L,U))
}

# Example 

A <- matrix(c(1,2,-3,-2,5,4,7,2,11), nrow=3, byrow=TRUE)
LU <- factorizeLU(A)
L<-LU[[1]]  
U<-LU[[2]]

A

##      [,1] [,2] [,3]
## [1,]    1    2   -3
## [2,]   -2    5    4
## [3,]    7    2   11

L

##      [,1]      [,2] [,3]
## [1,]    1  0.000000    0
## [2,]   -2  1.000000    0
## [3,]    7 -1.333333    1

U

##      [,1] [,2]     [,3]
## [1,]    1    2 -3.00000
## [2,]    0    9 -2.00000
## [3,]    0    0 29.33333

A == L %*% U

##      [,1] [,2] [,3]
## [1,] TRUE TRUE TRUE
## [2,] TRUE TRUE TRUE
## [3,] TRUE TRUE TRUE