CUNY 605

Week 2 HW Assignment

Problem Set 1

1. A^TA does not equal AA^T

Assume the following two matrices:

\[\mathbf{A} = \left[\begin{array} {rrr} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right] \]

\[\mathbf{A^T} = \left[\begin{array} {rrr} A_{11} & A_{21} \\ A_{12} & A_{22} \end{array}\right] \] If we multiply AA^T the value in the first cell (position 1,1) would be \[ A_{11}A_{11}+A_{12}A_{12} \] whereas if we multiply A^TA the value in the first cell would be \[A_{11}A_{11}+A_{21}A_{21}\] So clearly the two are not equivalent.

2. A^TA = AA^T only when A is symmetric. Using the above proof as an example, A₁₂ would be the same as A₂₁ if the matrix were symmetric.

Problem Set 2

LU_decomp <- function(A){
  
  ### Check to make sure matrix is not singular, rectangular, or comprised of zeros on pivot
  if(sum(diag(A)==0)>0 | qr(A)$rank<max(nrow(A), ncol(A))) {
    message("Error: This matrix cannot be decomposed.")
  } else {
    U <- A
    # Get U by going column by column and algebraically putting zeros in all positions below the diagonal
    for (j in 1:(ncol(U)-1)){
      for (i in (j+1):nrow(U)){
        if (U[i,j]!=0){U[i,] <- U[i, ] - U[j, ]*(U[i,j]/U[j,j])}
      }
    }
    
    # Get L by restating L U = A  as  U_inv L_inv = A_inv  ,solving for L_inv, and then taking inverse to get L 
    U_inv <- solve(U)
    A_inv <- solve(A)
    L_inv <- solve(U_inv, A_inv)
    L <- solve(L_inv)
    list(L=round(L,2) , U=round(U,2))
  }
}

### Test the function
test_A <- rbind(c(1, 1, 3),
                c(2, -1, 5),
                c(-1, -2, 4))
LU <- LU_decomp(test_A)

\[A= \begin{bmatrix} 1 & 1 & 3 \\ 2 & -1 & 5 \\ -1 & -2 & 4 \\ \end{bmatrix} \]

\[L= \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ -1 & 0.33 & 1 \\ \end{bmatrix} \]

\[U= \begin{bmatrix} 1 & 1 & 3 \\ 0 & -3 & -1 \\ 0 & 0 & 7.33 \\ \end{bmatrix} \]

# try matrix with zero pivot
test_A <- rbind(c(0,1,1),
                c(1,2,3),
                c(7,7,9))
LU_decomp(test_A)

## Error: This matrix cannot be decomposed.

# try matrix that is singular
test_A <- rbind(c(1,1,1),
                c(1,2,3),
                c(7,7,7))
LU_decomp(test_A)

## Error: This matrix cannot be decomposed.