1) Prove for that in general: \[A^TA \neq AA^T \]

Let the the transpose of A be equal to B for the sake of clarity.

\[ A^T = B\] So I am to prove \[ BA \neq AB \]

\[ AB = \left(\begin{array}{ccc} a_{11} & a_{12}\\ a_{21} & a_{22} \\ \end{array}\right) * \left(\begin{array}{cc} b_{11} & b_{12}\\ b_{21} & b_{22} \\ \end{array}\right) = \left(\begin{array}{ccc} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22}\\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} \\ \end{array}\right) \] \[ BA = \left(\begin{array}{cc} b_{11} & b_{12}\\ b_{21} & b_{22} \\ \end{array}\right)* \left(\begin{array}{ccc} a_{11} & a_{12}\\ a_{21} & a_{22} \\ \end{array}\right) = \left(\begin{array}{ccc} b_{11}a_{11}+b_{12}a_{21} & b_{11}a_{12}+b_{12}a_{22}\\ b_{21}a_{11}+b_{22}a_{21} & b_{21}a_{12}+b_{22}a_{22} \\ \end{array}\right) \]

There are special cases, such as the zero matrix; or any symmetrical matrix in such that \[A^T = A\] in which \[A^TA = AA^T \]

Question 2 : Write an R function to factorize a square matrix A into LDU

LU = function(A){
  
  #Check if square
  
  if(dim(A)[1]!=dim(A)[2]){
    return('Not a square matrix')
  }
  
  # Gather matrix dimensions
  
  rows = columns = dim(A)[1]
  
  
  
  # A = LDU
  # L = Lower triangle matrix 
  # D = Diagonal matrix 
  # U = Upper triangle matrix 
  
  U = A
  L = D =  diag(rows)
 
  #Column Loop
  for (j in 1:(columns-1)){
    #row loop
    for (i in (j+1):rows){
      #elimination
      L[i,j] = (U[i,j]/U[j,j])
      U[i,] = U[i,]-(U[j,]*L[i,j])
    }
  }
  #transfer the middle diagonal from upper triangular matrix
  diag(D) = diag(U)
  for (l in 1:rows){
    U[l,] = U[l,]/U[l,l]
  }
  
  LDU = list("Lower"=L,"Diagonal"=D,"Upper"=U)
return(LDU)
  
}
#The matrix I used in assignment 1
A = matrix(c(1,2,-1,1,-1,-2,3,5,4),nrow=3,ncol = 3)
LU(A)
## $Lower
##      [,1]      [,2] [,3]
## [1,]    1 0.0000000    0
## [2,]    2 1.0000000    0
## [3,]   -1 0.3333333    1
## 
## $Diagonal
##      [,1] [,2]     [,3]
## [1,]    1    0 0.000000
## [2,]    0   -3 0.000000
## [3,]    0    0 7.333333
## 
## $Upper
##      [,1] [,2]      [,3]
## [1,]    1    1 3.0000000
## [2,]    0    1 0.3333333
## [3,]    0    0 1.0000000

Sources [https://www.youtube.com/watch?v=rhNKncraJMk http://rstudio-pubs-static.s3.amazonaws.com/223289_e12b9066dfa24a38b354deacf591f118.html] http://54.225.166.221/simonnyc/107423