Akula-DATA605-Week02-HW2
Pavan Akula
September 9, 2017
A: (1)
Consider a matrix A with i rows and j columns,
\(A_{(i,j)} = \left[\begin{array}{cc}a_{1,1} & a_{1,2} & a_{1,3} & ..... & a_{1,j} \\a_{2,1} & a_{2,2} & a_{2,3} & ..... & a_{2,j} \\a_{3,1} & a_{3,2} & a_{3,3} & ..... & a_{3,j} \\. & . & . & ..... & .\\. & . & . & ..... & .\\. & . & . & ..... & .\\a_{i,1} & a_{i,2} & a_{i,3} & ..... & a_{i,j}\end{array}\right]\)
Lets say values, \(i \ne j\) and so on ….
\(A_{(j,i)}^{T}\) will result in j rows and i columns.
Transpose of A, \(A^T = \left[\begin{array}{cc}a_{1,1} & a_{2,1} & a_{3,1} & ..... & a_{i,1} \\a_{1,2} & a_{2,2} & a_{3,2} & ..... & a_{i,2} \\a_{1,3} & a_{2,3} & a_{3,3} & ..... & a_{i,3} \\. & . & . & ..... & .\\. & . & . & ..... & .\\. & . & . & ..... & .\\a_{1,j} & a_{2,j} & a_{3,j} & ..... & a_{j,i}\end{array}\right]\)
\(A_{(i,j)}A_{(j,i)}^{T}\) will result in matrix of i rows and i columns, \(AA_{(i,i)}^{T}\)
\(A_{(j,i)}^{T}A_{(i,j)}\) will result in matrix of j rows and j columns, \(A^{T}A_{(j,j)}\)
Hence, \(A_{(i,j)}A_{(j,i)}^{T} \ne A_{(j,i)}^{T}A_{(i,j)}\)
\(AA_{(i,i)}^{T} \ne A^{T}A_{(j,j)}\)
Example: Consider 4 X 5 matrix
\(A = \left[\begin{array}{cc}1 & 1 & 1 & 1 & 3 \\3 & 2 & 1 & 2 & 4 \\2 & 1 & 0 & 2 & 1 \\1 & 1 & 1 & 1 & 3\end{array}\right]\)
#4 X 5 matrix
A = matrix(c(1,3,2,1, 1,2,1,1, 1,1,0,1, 1,2,2,1, 3,4,1,3), nrow=4, ncol=5)
A## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 1 1 1 3
## [2,] 3 2 1 2 4
## [3,] 2 1 0 2 1
## [4,] 1 1 1 1 3#transpose of matrix A, generates 5 X 4
tA = t(A)
tA## [,1] [,2] [,3] [,4]
## [1,] 1 3 2 1
## [2,] 1 2 1 1
## [3,] 1 1 0 1
## [4,] 1 2 2 1
## [5,] 3 4 1 3#generates 4 X 4 matrix
A%*%tA## [,1] [,2] [,3] [,4]
## [1,] 13 20 8 13
## [2,] 20 34 16 20
## [3,] 8 16 10 8
## [4,] 13 20 8 13#generates 5 X 5 matrix
tA%*%A## [,1] [,2] [,3] [,4] [,5]
## [1,] 15 10 5 12 20
## [2,] 10 7 4 8 15
## [3,] 5 4 3 4 10
## [4,] 12 8 4 10 16
## [5,] 20 15 10 16 35Outputs are different matrices with different values.
A: (2)
When a matrix is symmetrical square matrix or identity matrix then \(A_{(i,j)}A_{(j,i)}^{T} = A_{(j,i)}^{T}A_{(i,j)}\)
#4 X 4 matrix
A = matrix(c(9,1,3,6, 1,11,7,6, 3,7,4,1, 6,6,1,10), nrow=4, ncol=4)
A## [,1] [,2] [,3] [,4]
## [1,] 9 1 3 6
## [2,] 1 11 7 6
## [3,] 3 7 4 1
## [4,] 6 6 1 10#transpose of matrix A
#it is similar to A
tA = t(A)
tA## [,1] [,2] [,3] [,4]
## [1,] 9 1 3 6
## [2,] 1 11 7 6
## [3,] 3 7 4 1
## [4,] 6 6 1 10#matrix multiplication
A%*%tA## [,1] [,2] [,3] [,4]
## [1,] 127 77 52 123
## [2,] 77 207 114 139
## [3,] 52 114 75 74
## [4,] 123 139 74 173tA%*%A## [,1] [,2] [,3] [,4]
## [1,] 127 77 52 123
## [2,] 77 207 114 139
## [3,] 52 114 75 74
## [4,] 123 139 74 173Symmetric Square Matrix is one exception to the rule, where (i = j),
\(A_{(i,j)}A_{(j,i)}^{T} = A_{(j,i)}^{T}A_{(i,j)}\) = \(AA_{(i,i)}^{T} = A^{T}A_{(j,j)}\)
Similarly, for identity matrix
#4 X 4 identity matrix
A = diag(4)
A## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0
## [2,] 0 1 0 0
## [3,] 0 0 1 0
## [4,] 0 0 0 1#transpose of matrix A
#it is similar to A
tA = t(A)
tA## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0
## [2,] 0 1 0 0
## [3,] 0 0 1 0
## [4,] 0 0 0 1#matrix multiplication
A%*%tA## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0
## [2,] 0 1 0 0
## [3,] 0 0 1 0
## [4,] 0 0 0 1tA%*%A## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0
## [2,] 0 1 0 0
## [3,] 0 0 1 0
## [4,] 0 0 0 1Identity matrix is another exception to the rule, where (i = j),
\(A_{(i,j)}A_{(j,i)}^{T} = A_{(j,i)}^{T}A_{(i,j)}\) = \(AA_{(i,i)}^{T} = A^{T}A_{(j,j)}\)
A:
Following function accepts a 5 X 5 square matrix and diagonal value to create identity matrix. It returns two matrices lower(L) and upper(U) decomposition.
Function is coded based on https://www.youtube.com/watch?v=UlWcofkUDDU.
lud_function <- function(a,dia){
#initiate identity matrix
l = diag(dia)
u = a
#check if a[1,1] equal zero move it following row
for (i in 1:nrow(u)){
if (u[1,1] == 0){
if (i+1 <= nrow(u)){
swap = u[i+1,]
u[i+1,] = u[1,]
u[1,] = swap
if (u[1,1] != 0){
break
}
}
}
}
#Loop through every column and every row.
#While working with column number 1 row number should be 2 (row > column)
#For the first row, leave values as is
#For the second row make first value zero
#For the third row make first and second column values zero
#Basically, we are doing row reduction echelon form, converting numbers on the lower side of diagonal to zero.
#And saving factor value on lower side of identity matrix.
k = 0
#read columns
for (j in 1:ncol(a)){
k = k + 1
#read rows
for (i in 1:nrow(u)){
#if row > column and value for the row is not equal to zero
if (i > j){
if (u[i,j] != 0){
#initiate the sign of the factor
s = -1
if (u[i,j] < 0){
if (u[k,j] < 0){
#when both numbers are negative keep sign positive
s = 1
}
}
#calculate the factor
f = s * u[i,j]/u[k,j]
#multiply upper row with the factor and sign
swap1 = s * (u[k,] * u[i,j])/u[k,j]
#get current row
swap2 = u[i,]
#add modified upper row to current row.
#this should make row,column value to zero when addition is performed
u[i,] = swap1 + swap2
#record the factor
l[i,j] = -1*f
}
}
}
}
return(list(l=l,u=u))
}
#get matrix
#a = matrix(c(1,-2,3, 4,8,4, -3,5,7), nrow=3, ncol=3)
#l = diag(3)
#a = matrix(c(1,2,0,0, 5,12,4,0, 0,5,13,6, 0,0,5,11), nrow=4, ncol=4)
#l = diag(4)
a = matrix(c(2,6,6,4,2, 3,1,3,2,1, 4,3,1,4,2, 1,1,2,7,4, 3,2,5,8,2), nrow=5, ncol=5)
m = lud_function(a,5)
#Lower matrix
m$l## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 0.00 0.0000000 0.000000 0
## [2,] 3 1.00 0.0000000 0.000000 0
## [3,] 3 -0.75 1.0000000 0.000000 0
## [4,] 2 -0.50 -0.4788732 1.000000 0
## [5,] 1 -0.25 -0.2394366 0.678392 1#Upper matrix
m$u## [,1] [,2] [,3] [,4] [,5]
## [1,] 2 3.000000 4.000000 1.000000e+00 3.0000000
## [2,] 0 -8.000000 -9.000000 -2.000000e+00 -7.0000000
## [3,] 0 -12.000000 -17.750000 -2.500000e+00 -9.2500000
## [4,] 0 -13.746479 -17.000000 2.802817e+00 -5.9295775
## [5,] 0 2.452261 3.032663 -2.220446e-16 -0.9422111Matrix multipication results in original matrix
#original matrix
a## [,1] [,2] [,3] [,4] [,5]
## [1,] 2 3 4 1 3
## [2,] 6 1 3 1 2
## [3,] 6 3 1 2 5
## [4,] 4 2 4 7 8
## [5,] 2 1 2 4 2#multipication of L and U
m$l%*%m$u## [,1] [,2] [,3] [,4] [,5]
## [1,] 2 3 4 1 3
## [2,] 6 1 3 1 2
## [3,] 6 3 1 2 5
## [4,] 4 2 4 7 8
## [5,] 2 1 2 4 2Both outputs match.