IS - 605 Assignment 02

1. Problem set 1

1- Show that transpose(A) A != A transpose(A) in general.

To prove the above, we simply provide an example that proves the above as below:

1. Solution

# creating a square matrix 3x3 A, as per below: 
A<- matrix(c(1,2,3,4,5,6, 7,8,9), nrow=3, ncol=3, byrow=TRUE)
A

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    4    5    6
## [3,]    7    8    9

# finding the transpose of A using t()
T<- t(A)
T

##      [,1] [,2] [,3]
## [1,]    1    4    7
## [2,]    2    5    8
## [3,]    3    6    9

# Comparing 

  if ((all.equal(A%*%T, T%*%A)) ==TRUE) print (" A%*%T is equal T%*%A ") else print(" A%*%T is not equal T%*%A ")

## [1] " A%*%T is not equal T%*%A "

(A%*%T == T%*%A)

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

# Hence there exist a Matrix A above where transpose(A) A <> A transpose(A) 


# 2- For a special type of square matrix A, we get AT A = AAT. Under what conditions
# could this be true? (Hint: The Identity matrix I is an example of such a matrix).
#

# The identity square matrix is special matrix where transpose(A) A == A transpose(A) 

A<- matrix(c(1,0,0,0,1,0,0,0,1), nrow=3, ncol=3, byrow=TRUE)
A

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

T<- t(A)
T

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

if ((all.equal(A%*%T, T%*%A)) ==TRUE) print (" A%*%T is equal T%*%A ") else print(" A%*%T is not equal T%*%A ")

## [1] " A%*%T is equal T%*%A "

(A%*%T == T%*%A)

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

# In addition, any square matrix where all elements are identical, the transpose(A) A == A transpose(A) holds true
# please note the below examples

A<- matrix(c(0,0,0,0,0,0,0,0,0), nrow=3, ncol=3, byrow=TRUE)
A

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

T<- t(A)
T

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

if ((all.equal(A%*%T, T%*%A)) ==TRUE) print (" A%*%T is equal T%*%A ") else print(" A%*%T is not equal T%*%A ")

## [1] " A%*%T is equal T%*%A "

(A%*%T == T%*%A)

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

##
A<- matrix(c(1,1,1,1,1,1,1,1,1), nrow=3, ncol=3, byrow=TRUE)
A

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

T<- t(A)
T

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

if ((all.equal(A%*%T, T%*%A)) ==TRUE) print (" A%*%T is equal T%*%A ") else print(" A%*%T is not equal T%*%A ")

## [1] " A%*%T is equal T%*%A "

(A%*%T == T%*%A)

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

##

##

A<- matrix(c(12,12,12,12,12,12,12,12,12), nrow=3, ncol=3, byrow=TRUE)
A

##      [,1] [,2] [,3]
## [1,]   12   12   12
## [2,]   12   12   12
## [3,]   12   12   12

T<- t(A)
T

##      [,1] [,2] [,3]
## [1,]   12   12   12
## [2,]   12   12   12
## [3,]   12   12   12

if ((all.equal(A%*%T, T%*%A)) ==TRUE) print (" A%*%T is equal T%*%A ") else print(" A%*%T is not equal T%*%A ")

## [1] " A%*%T is equal T%*%A "

(A%*%T == T%*%A)

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

2. Problem set 2

Write an R function to factorize a square matrix A into LU or LDU, whichever you

prefer. Please submit your response in an R Markdown document using our class naming

convention, E.g. GIyengar Assignment2 PS2.Rmd

You don’t have to worry about permuting rows of A and you can assume that A is less

than 5x5, if you need to hard-code any variables in your code. If you doing the entire

assignment in R, then please submit only one markdown document for both the problems

2- Solution

The function myfactorize(A) takes any square size matrix ,A ,as input. Then, it produces and prints the lower matrix L and

the upper matrix U.Then it calculates and prints the dot product L% * %U. Then it compare L% * %U to the original matrix A.

If the comparison is successful, it returns TRUE.

#####
myfactorize  <- function(A) {
  
  #  Array size
  d =dim(A)[1]
  d
  
  # create an identity matrix
  y <- diag( dim(A)[1])
  y
  
  # initializing parameters 
  
  temp = y
  temp[2,1] <- -A[2,1]/A[1,1]
  ltemp= temp
  
  # initialize the Upper elimination matrix 
  temp= temp %*% A
  L=ltemp
  i=1
  
  # initialize the lower elimination matrix
  L= solve(L)

   # looping through the martix columns  
  for  (j in 1:(d-i)) {
    #y <- diag( dim(A)[1])
    for  (i in 3:d) {
      # check if the pivot element. no permutaion is done when pivot is 0. 
      if (i != j) {
        # Check if the element is zero
        if (temp[i,j] != 0 ) {  
          if (temp[j,j] < 0)  y[i,j] <- -temp[i,j] / temp[j,j]     
          if (temp[j,j] > 0  )  y[i,j] <- -temp[i,j]/ temp[j,j]
          
          #place holder for the Uppper matrix U
          temp= y%*%temp
          
          # place holder for the Lower matrix L
          L= L %*%solve(y)
          
          #print(temp). Please uncomment to see all the Upper elimination matrices
          #print(L). Please uncomment to see all the Lower elimination matrices

          #reset y to the identity matrix
          y <- diag( dim(A)[1])
          
        }
        
      }
      
    }
  } 
  print("Upper Matrix U")
  print(temp)
  print("Lower Matrix L")
  print(L)
  U=temp
  solution <- L%*%U
  print("Compare L%*%U with original Matrix A. Below is the L%*%U matrix:  ")
  print (solution) 
  print("Below is the original Matrix A: ")
  print(A)
  return(all.equal(solution,A))  
  
}

Testing

Please note that to see all the steps/ elimination matrices, print(temp) and print(L) statements need to be uncommented

They were commented for ease readability. They were occasionally commented for debugging.

Also, please note that the testing below was tested against the online matrix factorization below. it only goes up to 10x10.

http://www.gregthatcher.com/Mathematics/LU_Factorization.aspx

#  Testing for 3x3 Matrix. Note: The below 3x3 matrix is we have in the course module 2
A <- matrix(c(1,2,3,1,1,1,2,0,1),nrow=3, byrow=TRUE)
A

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

myfactorize(A)

## [1] "Upper Matrix U"
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0   -1   -2
## [3,]    0    0    3
## [1] "Lower Matrix L"
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    1    1    0
## [3,]    2    4    1
## [1] "Compare L%*%U with original Matrix A. Below is the L%*%U matrix:  "
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    1    1    1
## [3,]    2    0    1
## [1] "Below is the original Matrix A: "
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    1    1    1
## [3,]    2    0    1

## [1] TRUE

# Testing a 4x4 matrix  
A<- matrix(c(1,2,3,4,4.0,3,2,1,-5,-6,7,8,-8,-7,-6,5),nrow=4,byrow=TRUE)
A

##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4
## [2,]    4    3    2    1
## [3,]   -5   -6    7    8
## [4,]   -8   -7   -6    5

myfactorize(A)

## [1] "Upper Matrix U"
##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4
## [2,]    0   -5  -10  -15
## [3,]    0    0   14   16
## [4,]    0    0    0   10
## [1] "Lower Matrix L"
##      [,1] [,2] [,3] [,4]
## [1,]    1  0.0    0    0
## [2,]    4  1.0    0    0
## [3,]   -5 -0.8    1    0
## [4,]   -8 -1.8    0    1
## [1] "Compare L%*%U with original Matrix A. Below is the L%*%U matrix:  "
##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4
## [2,]    4    3    2    1
## [3,]   -5   -6    7    8
## [4,]   -8   -7   -6    5
## [1] "Below is the original Matrix A: "
##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4
## [2,]    4    3    2    1
## [3,]   -5   -6    7    8
## [4,]   -8   -7   -6    5

## [1] TRUE

# Testing for a 5x5 matrix. Please note that in the example, a row permutation step was omitted 
# specifically row 3 and row 5. However, a validation checks of L%*%U == A resulted into successful test.


A<- matrix(c(1,2,3,4,5,5,4,3,2,1,6,7,8,9,1,1,2,2,1,3,3,3,1,6,1), nrow=5,byrow=TRUE)
A

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    2    3    4    5
## [2,]    5    4    3    2    1
## [3,]    6    7    8    9    1
## [4,]    1    2    2    1    3
## [5,]    3    3    1    6    1

myfactorize(A)

## [1] "Upper Matrix U"
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    2    3    4    5
## [2,]    0   -6  -12  -18  -24
## [3,]    0    0    0    0   -9
## [4,]    0    0   -1   -3   -2
## [5,]    0    0   -3    0   -4
## [1] "Lower Matrix L"
##      [,1]      [,2] [,3] [,4] [,5]
## [1,]    1 0.0000000    0    0    0
## [2,]    5 1.0000000    0    0    0
## [3,]    6 0.8333333    1    0    0
## [4,]    1 0.0000000    0    1    0
## [5,]    3 0.5000000    0   -1    1
## [1] "Compare L%*%U with original Matrix A. Below is the L%*%U matrix:  "
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    2    3    4    5
## [2,]    5    4    3    2    1
## [3,]    6    7    8    9    1
## [4,]    1    2    2    1    3
## [5,]    3    3    1    6    1
## [1] "Below is the original Matrix A: "
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    2    3    4    5
## [2,]    5    4    3    2    1
## [3,]    6    7    8    9    1
## [4,]    1    2    2    1    3
## [5,]    3    3    1    6    1

## [1] TRUE

#
# Testing for a 6x6 matrix. 
A <- matrix(c(1,    7/2,    2,  6,  1,  7/2,
              2,    9/2,    9,  7/4,    9,  1,
              1/4,  1,  1,  7/3,    2,  3,
              8,    2/3,    5,  2,  8,  6,
              7,    6,  2,  3,  6,  3,
              3/2,  5,  2,  3,  7,  4/3),nrow=6, byrow=TRUE)
A

##      [,1]      [,2] [,3]     [,4] [,5]     [,6]
## [1,] 1.00 3.5000000    2 6.000000    1 3.500000
## [2,] 2.00 4.5000000    9 1.750000    9 1.000000
## [3,] 0.25 1.0000000    1 2.333333    2 3.000000
## [4,] 8.00 0.6666667    5 2.000000    8 6.000000
## [5,] 7.00 6.0000000    2 3.000000    6 3.000000
## [6,] 1.50 5.0000000    2 3.000000    7 1.333333

myfactorize(A)

## [1] "Upper Matrix U"
##      [,1]          [,2] [,3]          [,4]     [,5]        [,6]
## [1,]    1  3.500000e+00 2.00  6.000000e+00  1.00000   3.5000000
## [2,]    0 -2.500000e+00 5.00 -1.025000e+01  7.00000  -6.0000000
## [3,]    0  1.923899e-16 0.75 -6.661267e-16  0.00000  -0.4857514
## [4,]    0  1.620886e-14 0.00  9.415741e+01  0.00000 103.2687687
## [5,]    0 -2.181308e-15 0.00  7.105427e-15 18.49909  17.2558757
## [6,]    0  1.807197e-15 0.00 -5.354192e-15  0.00000  -3.3091736
## [1] "Lower Matrix L"
##      [,1]     [,2]      [,3]         [,4]       [,5] [,6]
## [1,] 1.00  0.00000   0.00000  0.000000000  0.0000000    0
## [2,] 2.00  1.00000   0.00000  0.000000000  0.0000000    0
## [3,] 0.25 -0.05000   1.00000  0.003407415  0.1135191    0
## [4,] 8.00 10.93333 -87.55556  0.701661914 -4.1371415    0
## [5,] 7.00  7.40000 -65.33333  0.391365916 -2.8541952    0
## [6,] 1.50  0.10000  -2.00000 -0.052837054  0.2594723    1
## [1] "Compare L%*%U with original Matrix A. Below is the L%*%U matrix:  "
##      [,1]      [,2] [,3]     [,4] [,5]     [,6]
## [1,] 1.00 3.5000000    2 6.000000    1 3.500000
## [2,] 2.00 4.5000000    9 1.750000    9 1.000000
## [3,] 0.25 1.0000000    1 2.333333    2 3.000000
## [4,] 8.00 0.6666667    5 2.000000    8 6.000000
## [5,] 7.00 6.0000000    2 3.000000    6 3.000000
## [6,] 1.50 5.0000000    2 3.000000    7 1.333333
## [1] "Below is the original Matrix A: "
##      [,1]      [,2] [,3]     [,4] [,5]     [,6]
## [1,] 1.00 3.5000000    2 6.000000    1 3.500000
## [2,] 2.00 4.5000000    9 1.750000    9 1.000000
## [3,] 0.25 1.0000000    1 2.333333    2 3.000000
## [4,] 8.00 0.6666667    5 2.000000    8 6.000000
## [5,] 7.00 6.0000000    2 3.000000    6 3.000000
## [6,] 1.50 5.0000000    2 3.000000    7 1.333333

## [1] TRUE

# Testing for a 10x10 matrix. 
#
A<- matrix(c(1, 3,  2,  3,  5,  2/3,    7,  3,  3/2,    8,
             9/4,   3/2,    2,  1/2,    9,  2,  2,  8/3,    6,  2,
             9, 2,  4,  8,  9/4,    9,  9/4,    2,  8,  1/3,
             1, 2,  4,  6,  7,  8,  2,  2/3,    4,  5,
             7/2,   2,  7/3,    3,  1,  5,  9,  6,  5/3,    3,
             6, 4/3,    1,  1,  5,  2,  1,  6,  7/3,    5,
             2/3,   3,  8,  1,  1,  2,  7,  1/3,    7,  1,
             2, 8/3,    5,  6,  5/4,    4/3,    1/3,    5,  5,  9/2,
             7, 7/4,    3,  5/4,    8,  4,  9/4,    9/4,    1,  6,
             3, 6,  1,  3,  7,  9,  5,  9/2,    2,  9), nrow=10, byrow=TRUE)

A

##            [,1]     [,2]     [,3] [,4] [,5]      [,6]      [,7]      [,8]
##  [1,] 1.0000000 3.000000 2.000000 3.00 5.00 0.6666667 7.0000000 3.0000000
##  [2,] 2.2500000 1.500000 2.000000 0.50 9.00 2.0000000 2.0000000 2.6666667
##  [3,] 9.0000000 2.000000 4.000000 8.00 2.25 9.0000000 2.2500000 2.0000000
##  [4,] 1.0000000 2.000000 4.000000 6.00 7.00 8.0000000 2.0000000 0.6666667
##  [5,] 3.5000000 2.000000 2.333333 3.00 1.00 5.0000000 9.0000000 6.0000000
##  [6,] 6.0000000 1.333333 1.000000 1.00 5.00 2.0000000 1.0000000 6.0000000
##  [7,] 0.6666667 3.000000 8.000000 1.00 1.00 2.0000000 7.0000000 0.3333333
##  [8,] 2.0000000 2.666667 5.000000 6.00 1.25 1.3333333 0.3333333 5.0000000
##  [9,] 7.0000000 1.750000 3.000000 1.25 8.00 4.0000000 2.2500000 2.2500000
## [10,] 3.0000000 6.000000 1.000000 3.00 7.00 9.0000000 5.0000000 4.5000000
##           [,9]     [,10]
##  [1,] 1.500000 8.0000000
##  [2,] 6.000000 2.0000000
##  [3,] 8.000000 0.3333333
##  [4,] 4.000000 5.0000000
##  [5,] 1.666667 3.0000000
##  [6,] 2.333333 5.0000000
##  [7,] 7.000000 1.0000000
##  [8,] 5.000000 4.5000000
##  [9,] 1.000000 6.0000000
## [10,] 2.000000 9.0000000

myfactorize(A)

## [1] "Upper Matrix U"
##       [,1]          [,2]      [,3]          [,4]      [,5]      [,6]
##  [1,]    1  3.000000e+00  2.000000  3.000000e+00  5.000000 0.6666667
##  [2,]    0 -5.250000e+00 -2.500000 -6.250000e+00 -2.250000 0.5000000
##  [3,]    0 -2.967045e-16 -2.095238  2.373636e-15  0.000000 0.0000000
##  [4,]    0  2.413348e-15  0.000000  1.690909e+01  0.000000 0.0000000
##  [5,]    0 -2.190619e-16  0.000000  1.752495e-15 -4.566756 0.0000000
##  [6,]    0 -2.860596e-16  0.000000  2.288477e-15  0.000000 2.4187514
##  [7,]    0 -1.326042e-15  0.000000  1.060834e-14  0.000000 0.0000000
##  [8,]    0  4.032790e-16  0.000000 -3.226232e-15  0.000000 0.0000000
##  [9,]    0  1.453145e-16  0.000000 -1.162516e-15  0.000000 0.0000000
## [10,]    0  2.346907e-15  0.000000 -2.232797e-14  0.000000 0.0000000
##            [,7]          [,8]      [,9]       [,10]
##  [1,]   7.00000  3.000000e+00  1.500000   8.0000000
##  [2,] -13.75000 -4.083333e+00  2.625000 -16.0000000
##  [3,]   0.00000  0.000000e+00  0.000000   2.2331876
##  [4,]   0.00000  0.000000e+00  0.000000 -14.8818179
##  [5,]   0.00000  0.000000e+00  0.000000   0.5288498
##  [6,]   0.00000  0.000000e+00  0.000000   2.0195232
##  [7,] -28.64815  1.776357e-15  0.000000  25.5519258
##  [8,]   0.00000  1.067082e+01  0.000000   1.0078704
##  [9,]   0.00000  0.000000e+00 -7.228804   4.9298986
## [10,]   0.00000  0.000000e+00  0.000000 -27.9707877
## [1] "Lower Matrix L"
##            [,1]       [,2]       [,3]       [,4]       [,5]       [,6]
##  [1,] 1.0000000  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000
##  [2,] 2.2500000  1.0000000  0.0000000  0.0000000  0.0000000  0.0000000
##  [3,] 9.0000000  4.7619048  1.0000000  0.6364567  7.0149823  0.2559368
##  [4,] 1.0000000  0.1904762 -1.1818182  0.2478239 -0.5317935  2.9924924
##  [5,] 3.5000000  1.6190476  0.2954545  0.1548899  2.8153775  0.7678105
##  [6,] 6.0000000  3.1746032  1.4621212  0.1680321  3.9102465 -1.4831212
##  [7,] 0.6666667 -0.1904762 -2.9545455 -0.1295443  0.6047848  0.6824983
##  [8,] 2.0000000  0.6349206 -1.2348485  0.2346817  1.6032011 -0.1312497
##  [9,] 7.0000000  3.6666667  0.8750000  0.1872760  4.1057589 -1.0335911
## [10,] 3.0000000  0.5714286  1.7045455 -0.1436252  1.4702527  2.7759304
##              [,7]        [,8]        [,9] [,10]
##  [1,]  0.00000000  0.00000000  0.00000000     0
##  [2,]  0.00000000  0.00000000  0.00000000     0
##  [3,] -0.16497369 -0.52063051  2.49003830     0
##  [4,]  0.08311017 -0.14577654 -0.27667092     0
##  [5,] -0.23603289  0.19783959  1.08362778     0
##  [6,] -0.09252932  0.09024262  2.07503192     0
##  [7,]  0.00997322 -0.22907742 -0.89918050     0
##  [8,]  0.17231509  0.14924741 -0.04611182     0
##  [9,] -0.12798966 -0.35402875  2.64566569     0
## [10,]  0.28423678 -0.20304590  0.55334184     1
## [1] "Compare L%*%U with original Matrix A. Below is the L%*%U matrix:  "
##            [,1]     [,2]     [,3] [,4] [,5]      [,6]      [,7]      [,8]
##  [1,] 1.0000000 3.000000 2.000000 3.00 5.00 0.6666667 7.0000000 3.0000000
##  [2,] 2.2500000 1.500000 2.000000 0.50 9.00 2.0000000 2.0000000 2.6666667
##  [3,] 9.0000000 2.000000 4.000000 8.00 2.25 9.0000000 2.2500000 2.0000000
##  [4,] 1.0000000 2.000000 4.000000 6.00 7.00 8.0000000 2.0000000 0.6666667
##  [5,] 3.5000000 2.000000 2.333333 3.00 1.00 5.0000000 9.0000000 6.0000000
##  [6,] 6.0000000 1.333333 1.000000 1.00 5.00 2.0000000 1.0000000 6.0000000
##  [7,] 0.6666667 3.000000 8.000000 1.00 1.00 2.0000000 7.0000000 0.3333333
##  [8,] 2.0000000 2.666667 5.000000 6.00 1.25 1.3333333 0.3333333 5.0000000
##  [9,] 7.0000000 1.750000 3.000000 1.25 8.00 4.0000000 2.2500000 2.2500000
## [10,] 3.0000000 6.000000 1.000000 3.00 7.00 9.0000000 5.0000000 4.5000000
##           [,9]     [,10]
##  [1,] 1.500000 8.0000000
##  [2,] 6.000000 2.0000000
##  [3,] 8.000000 0.3333333
##  [4,] 4.000000 5.0000000
##  [5,] 1.666667 3.0000000
##  [6,] 2.333333 5.0000000
##  [7,] 7.000000 1.0000000
##  [8,] 5.000000 4.5000000
##  [9,] 1.000000 6.0000000
## [10,] 2.000000 9.0000000
## [1] "Below is the original Matrix A: "
##            [,1]     [,2]     [,3] [,4] [,5]      [,6]      [,7]      [,8]
##  [1,] 1.0000000 3.000000 2.000000 3.00 5.00 0.6666667 7.0000000 3.0000000
##  [2,] 2.2500000 1.500000 2.000000 0.50 9.00 2.0000000 2.0000000 2.6666667
##  [3,] 9.0000000 2.000000 4.000000 8.00 2.25 9.0000000 2.2500000 2.0000000
##  [4,] 1.0000000 2.000000 4.000000 6.00 7.00 8.0000000 2.0000000 0.6666667
##  [5,] 3.5000000 2.000000 2.333333 3.00 1.00 5.0000000 9.0000000 6.0000000
##  [6,] 6.0000000 1.333333 1.000000 1.00 5.00 2.0000000 1.0000000 6.0000000
##  [7,] 0.6666667 3.000000 8.000000 1.00 1.00 2.0000000 7.0000000 0.3333333
##  [8,] 2.0000000 2.666667 5.000000 6.00 1.25 1.3333333 0.3333333 5.0000000
##  [9,] 7.0000000 1.750000 3.000000 1.25 8.00 4.0000000 2.2500000 2.2500000
## [10,] 3.0000000 6.000000 1.000000 3.00 7.00 9.0000000 5.0000000 4.5000000
##           [,9]     [,10]
##  [1,] 1.500000 8.0000000
##  [2,] 6.000000 2.0000000
##  [3,] 8.000000 0.3333333
##  [4,] 4.000000 5.0000000
##  [5,] 1.666667 3.0000000
##  [6,] 2.333333 5.0000000
##  [7,] 7.000000 1.0000000
##  [8,] 5.000000 4.5000000
##  [9,] 1.000000 6.0000000
## [10,] 2.000000 9.0000000

## [1] TRUE

Note that the echo = FALSE parameter was added to the code chunk to prevent printing of the R code that generated the plot.

IS - 605 Assignment 02

Mohamed Elmoudni

September 6, 2015

1. Problem set 1

1- Show that transpose(A) A != A transpose(A) in general.

To prove the above, we simply provide an example that proves the above as below:

1. Solution

2. Problem set 2

Write an R function to factorize a square matrix A into LU or LDU, whichever you

prefer. Please submit your response in an R Markdown document using our class naming

convention, E.g. GIyengar Assignment2 PS2.Rmd

You don’t have to worry about permuting rows of A and you can assume that A is less

than 5x5, if you need to hard-code any variables in your code. If you doing the entire

assignment in R, then please submit only one markdown document for both the problems

2- Solution

The function myfactorize(A) takes any square size matrix ,A ,as input. Then, it produces and prints the lower matrix L and

the upper matrix U.Then it calculates and prints the dot product L% * %U. Then it compare L% * %U to the original matrix A.

If the comparison is successful, it returns TRUE.

Testing

Please note that to see all the steps/ elimination matrices, print(temp) and print(L) statements need to be uncommented

They were commented for ease readability. They were occasionally commented for debugging.

Also, please note that the testing below was tested against the online matrix factorization below. it only goes up to 10x10.

http://www.gregthatcher.com/Mathematics/LU_Factorization.aspx