DATA 605 - Assignment 4

Problem Set 1

Given a 3 × 2 matrix A write code in R to compute X = AA^T and Y = A^TA.

Matrix $A$

\[ A=\begin{bmatrix} 1 & 2 & 3 \\ -1 & 0 & 4 \end{bmatrix} \]

compute X = AA^T and Y = A^TA

A <- matrix(c(1, -1, 2, 0, 3, 4), 2)
X <- A %*% t(A)
Y <- t(A) %*% A

Matrix $X=AA^T$

##      [,1] [,2]
## [1,]   14   11
## [2,]   11   17

Matrix $Y=A^TA$

##      [,1] [,2] [,3]
## [1,]    2    2   -1
## [2,]    2    4    6
## [3,]   -1    6   25

Then, compute the eigenvalues and eigenvectors of X and Y using the built-in commans in R.

compute the eigenvalues and eigenvectors of X and Y

eigenX <- eigen(X)
eigenY <- eigen(Y)

Eigen Values for $X$

round(eigenX$values, digits=4)

## [1] 26.6018  4.3982

Eigen Values for $Y$

round(eigenY$values, digits=4)

## [1] 26.6018  4.3982  0.0000

Eigenvectors for $X$

round(eigenX$vectors, digits=4)

##        [,1]    [,2]
## [1,] 0.6576 -0.7534
## [2,] 0.7534  0.6576

Eigenvectors for $Y$

round(eigenY$vectors, digits=4)

##         [,1]    [,2]    [,3]
## [1,] -0.0186 -0.6728  0.7396
## [2,]  0.2550 -0.7185 -0.6472
## [3,]  0.9668  0.1766  0.1849

Then, compute the left-singular, singular values, and right-singular vectors of A using the svd command.

Calculate SVD of $A$

svdA <- svd(A)

Singular Values is $d below:

Left Vector is $u below:

Right Vector is $v below:

## $d
## [1] 5.157693 2.097188
## 
## $u
##            [,1]       [,2]
## [1,] -0.6576043 -0.7533635
## [2,] -0.7533635  0.6576043
## 
## $v
##             [,1]       [,2]
## [1,]  0.01856629 -0.6727903
## [2,] -0.25499937 -0.7184510
## [3,] -0.96676296  0.1765824

Examine the two sets of singular vectors and show that they are indeed eigenvectors of X and Y. In addition, the two non-zero eigenvalues (the 3rd value will be very close to zero, if not zero) of both X and Y are the same and are squares of the non-zero singular values of A.

round(svd(X)$d, digits=4)

## [1] 26.6018  4.3982

round(eigenX$values, digits=4)

## [1] 26.6018  4.3982

round(svd(Y)$d, digits=4)

## [1] 26.6018  4.3982  0.0000

round(eigenY$values, digits=4)

## [1] 26.6018  4.3982  0.0000

squares <- c(svd(A)$d)
squares <- (squares^2) 
squares

## [1] 26.601802  4.398198

Problem Set 2

A function to compute the inverse of a well-conditioned full-rank square matrix using co-factors.

myinverse <- function(A){
  matrix_of_cofactors <- matrix(NA, nrow = nrow(A), ncol = ncol(A)) # Initialize cofactors
   for (i in c(1:nrow(A))){    # outer loop
  
       for (j in c(1:ncol(A))){  # inner loop
      
      matrix_of_cofactors[i, j] <- det(A[-i, -j]) * (-1) ^ (i + j)   # Calc determinant of the matrix 
    }
  }
  
  adjugate <- t(matrix_of_cofactors)  # Transpose elements of the previous matrix of cofactors
  
  inverse <- adjugate / det(A)  # Divide ajugate by the determinant of the first matrix
  return(inverse)
}

The function works by computing the co-factor matrix using : $C_{ij} = (-1)^{i+j} det(M_{ij})$. The inverse matrix is then calculated using $A^{-1}=\frac{C^T}{det(A)}$.

Demonstration

Let $A=\begin{bmatrix} 1 & 6 & 7 \\2 & 5 & 8 \\ 3 & 6 & 9\end{bmatrix}$.

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

# Calculate inverse using 'myinverse' function
B <- myinverse(A)
B

##       [,1] [,2]       [,3]
## [1,] -0.25   -1  1.0833333
## [2,]  0.50   -1  0.5000000
## [3,] -0.25    1 -0.5833333

solve(A)

##       [,1] [,2]       [,3]
## [1,] -0.25   -1  1.0833333
## [2,]  0.50   -1  0.5000000
## [3,] -0.25    1 -0.5833333

round(solve(A), digits=10) == round(B, digits=10)

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

# Does AB=I
round(A %*% B, digits=10)

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

DATA 605 - Assignment 4

Jim Mundy

Problem Set 1

Matrix \(A\)

compute X = AA^T and Y = A^TA

Matrix \(X=AA^T\)

Matrix \(Y=A^TA\)

compute the eigenvalues and eigenvectors of X and Y

Eigen Values for \(X\)

Eigen Values for \(Y\)

Eigenvectors for \(X\)

Eigenvectors for \(Y\)

Calculate SVD of \(A\)

Singular Values is $d below:

Left Vector is $u below:

Right Vector is $v below:

Problem Set 2

Demonstration

DATA 605 - Assignment 4

Jim Mundy

Problem Set 1

Matrix \(A\)

compute X = AAT and Y = ATA

Matrix \(X=AA^T\)

Matrix \(Y=A^TA\)

compute the eigenvalues and eigenvectors of X and Y

Eigen Values for \(X\)

Eigen Values for \(Y\)

Eigenvectors for \(X\)

Eigenvectors for \(Y\)

Calculate SVD of \(A\)

Singular Values is $d below:

Left Vector is $u below:

Right Vector is $v below:

Problem Set 2

Demonstration

compute X = AA^T and Y = A^TA