Let’s assume that
\[A = \begin{pmatrix} a & b \\ c & d \\ e & f \\ \end{pmatrix}\]
The transpose of a matrix A is the matrix \(A^T\) whose rows are the columns of the matrix A.
\[A^T = \begin{pmatrix} a & c & e \\ b & d & f \\ \end{pmatrix}\]
For two matrices to be equal, they must be of the same size and have all the same entries.
If we multiply \(A^T\) by \(A\) we will get the matrix of the size 3x3. Whereas, if we multiply A by \(A^T\) we will get the matrix of the size 2x2.
So that, in order to be equal matrices \(A\) by \(A^T\) should have the same number of rows and columns. In other words the matrix A should have equal number of rows and numbers (be a squired matrix).
Let’s assume that the matrix A is squired.
\[A = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}\]
Now we can build the transpose of a matrix A.
\[A^T = \begin{pmatrix} a & c \\ b & d \\ \end{pmatrix}\]
\[A*A^T= \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) % \left( \begin{array}{cc} a & c \\ b & d \end{array} \right) = \left( \begin{array}{cc} a^2+b^2 & ac+bd \\ ac+bd & c^2+d^2 \end{array} \right)\]
\[A^T*A= \left( \begin{array}{cc} a & c \\ b & d \end{array} \right) % \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) = \left( \begin{array}{cc} a^2+c^2 & ab+cd \\ ab+dc & b^2+d^2 \end{array} \right)\]
Since corresponding elements of \(A^T\)\(*A\) and \(A*\)\(A^T\) are not equal we can conclude that in general \(A^T\)\(*A\) \(\neq\) \(A*\)\(A^T\).
Under what conditions could this be true? (Hint: The Identity matrix I is an example of such a matrix).
As we already figured out the matrix A should be a squired matrix (the number of rows should be equal the number of columns).
The first case when \(A^T\)\(*A\) = \(A*\)\(A^T\) is when \(A^T\) equals to \(A\). When \(A^T\)=\(A\) we have \(A\)\(*A\) = \(A*\)\(A\). Hence, \(A^2\) = \(A^2\).
I figured out that first row of A should be equal to first column, second row of A should be equal to contuse column, etc. So that, corresponding row should be equal to corresponding column.
Let’s create such a matrix
\[A = \begin{pmatrix} a & b & c \\ b & c & d \\ c & d & e \\ \end{pmatrix}\]
Then the transpose matrix equal to
\[A^T = \begin{pmatrix} a & b & c \\ b & c & d \\ c & d & e \\ \end{pmatrix}\]
As you can see \(A^T\)=\(A\).
The other case when \(A^T\)\(*A\) = \(A*\)\(A^T\) is when the expression \(A^T\)\(*A\) equals to identity matrix. If \(A^T\)\(*A\) equals o Identity matrix then \(A^T\) equals \(A^-1\). When \(A^T\) equals \(A^-1\) \(A^-1\)\(*A\) = I = \(A\)\(*A^-1\)
Matrix factorization is a very important problem. There are supercomputers built just to do matrix factorizations. Every second you are on an airplane, matrices are being factorized. Radars that track flights use a technique called Kalman filtering. At the heart of Kalman Filtering is a Matrix Factorization operation. Kalman Filters are solving linear systems of equations when they track your flight using radars. Write an R function to factorize a square matrix A into LU or LDU, whichever you prefer.
LU <- function(A,row_num,col_num) {
#create matrix L that consist of 0s
L = matrix(c(0:0),nrow=row_num, ncol=col_num, byrow = TRUE)
#assign 1s to the main diaginal
i=1
for (i in 1:3){
L[i,i]=1
}
#assign matrix A to initial matrix U
U <- A
print("matrix A")
print(A)
#start from the first column
col=1
#iterate until col_num-1
for (col in 1:(col_num-1)){
#start to iterate from the second row until final row
row=col+1
for (row in (col+1):row_num){
#U[row,col] = U[row,col]+x*U[col,col]
x=-U[row,col]/U[col,col]
U[row,col] = 0
#multiply the remaining elements of a current row by x
col2=col+1
for (col2 in (col+1):col_num){
U[row,col2] = U[row,col2]+x*U[col,col2]
print("matrix U")
print(U)
}
#assign x to a corresponding cell in matrix L
L[row,col] = -x
}
}
print("matrix L")
print(L)
#dot product of L and U should produce A
print("matrix LU=A")
print(L %*% U)
}Let’s test matrix 3x3
row_num <- 3
col_num <- row_num
A = matrix(c(1:(row_num*col_num)),nrow=row_num, ncol=col_num, byrow = TRUE)
LU(A,3,3)## [1] "matrix A"
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6
## [3,] 7 8 9
## [1] "matrix U"
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 0 -3 6
## [3,] 7 8 9
## [1] "matrix U"
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 0 -3 -6
## [3,] 7 8 9
## [1] "matrix U"
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 0 -3 -6
## [3,] 0 -6 9
## [1] "matrix U"
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 0 -3 -6
## [3,] 0 -6 -12
## [1] "matrix U"
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 0 -3 -6
## [3,] 0 0 0
## [1] "matrix L"
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 4 1 0
## [3,] 7 2 1
## [1] "matrix LU=A"
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6
## [3,] 7 8 9