#Problem Set 1 ## 1. Given Our Matrix
A <- matrix(c(2,5,3,5,0,1,0,3,3,2,1,2,2,4,2,1,5,6), nrow = 6)
A## [,1] [,2] [,3]
## [1,] 2 0 2
## [2,] 5 3 4
## [3,] 3 3 2
## [4,] 5 2 1
## [5,] 0 1 5
## [6,] 1 2 6A_t <- t(A)
A_t## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 2 5 3 5 0 1
## [2,] 0 3 3 2 1 2
## [3,] 2 4 2 1 5 6We see that our matrices of A and A_t are shown to haeb the dimensions 6x3 and 3x6 respectively
A_t%*%A## [,1] [,2] [,3]
## [1,] 64 36 41
## [2,] 36 27 37
## [3,] 41 37 86A%*%A_t## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 8 18 10 12 10 14
## [2,] 18 50 32 35 23 35
## [3,] 10 32 22 23 13 21
## [4,] 12 35 23 30 7 15
## [5,] 10 23 13 7 26 32
## [6,] 14 35 21 15 32 41When we do A_t*A what we are doing is taking the 3x6 matrix and multiplying it by the 6x3 matrix. This matches up as the rows from A_t match the columns from A. Thus the resulting matrix will have a resulting dimension : 3x3
However when we do A* A_t we take a 6x3 matrix and multiply it by a 3x6. While the rows and columns match up to allow the multiplication the dimensions are now : 6x6
The condition that A_tA = AA_t is only valid for a square matrix that is normal. A matrix is said to be normal if At = A. And to the prompt this could be an an example of the identity matrix
An example:
nmat <- matrix(c(1,0,0,0,2,0,0,0,3), nrow = 3)
nmat## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 2 0
## [3,] 0 0 3nmatt<- t(nmat)
nmatt## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 2 0
## [3,] 0 0 3nmatt%*%nmat## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 4 0
## [3,] 0 0 9nmat%*%nmatt## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 4 0
## [3,] 0 0 9The above matrix nmat is a diagonal matrix
mx <- matrix(c(1,2,3,4,5,6,7,8,9), nrow=3)
mx## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 2 5 8
## [3,] 3 6 9LU = function(x){
if (nrow(x) >=5){
return("Matrix dimensions larger than expected")
}
d = dim(x)
if (d[1] != d[2]){
return("Not a square matrix")
}
n = nrow(x)
l = diag(n)
f= 0 #initialize
while (f <= dim(x)[1]){ #While the first dimmension(row) of our input variable is less than or equal to our f which is initialized at 0
f = f + 1 #f here will represent a stopper for our function to stop looping
i = f #i here will represent the row
j = f #j will represent our columns
while (i <= dim(x)[1] -1) { # This line tells us that while our row is less than or equal to the rows of our input's row dimension -1
c = x[i + 1, j]/ x[j,j] # This code points to an exact coordinate of our input(*2,1), then it divides that by the coordinates of our current selected first one is *1,1
x[i+1,] = x[i+1,]- c*x[j,] #This code changes the input of our row +1 (*2nd row) to be the second row - our current row multiploied by our multiplier
l[i+1, j] = c #We now create our L matrix from our diagonal initialized. so we turn our input into upper and have our lower to return
i = 1+ i #We now increase i for the next row and repeat iteration
}
}
return(list(l,x))
}
LU(mx)## [[1]]
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 2 1 0
## [3,] 3 2 1
##
## [[2]]
## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 0 -3 -6
## [3,] 0 0 0library(pracma)## Warning: package 'pracma' was built under R version 4.2.2lu(mx)## $L
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 2 1 0
## [3,] 3 2 1
##
## $U
## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 0 -3 -6
## [3,] 0 0 0