Problem Set 1

Build matrices and dot products

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

Get eigen values with eigen command

y_eigen <- eigen(Y)
x_eigen <- eigen(X)
round(y_eigen$values,3)
## [1] 26.602  4.398  0.000
round(x_eigen$values,3)
## [1] 26.602  4.398

Build SVD

svd(A)
## $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

Comparing eigen values to singular values

  • The singular values of a \[X_{SV} = sqrt((X^T*X)_{eig})\]
sing_val_x <- svd(A)$d
round(sing_val_x,3)==round(sqrt(x_eigen$values),3)
## [1] TRUE TRUE

Comparing eigen vectors

  • our functions return a different standardized eigen vector below
my_svd <- svd(A)

x_eigen$vectors==my_svd$u
##       [,1] [,2]
## [1,] FALSE TRUE
## [2,] FALSE TRUE
  • link2
  • multiplying by a constant of -1 will still give us a valid eigenvector
my_svd$u[,1] <- my_svd$u[,1]*-1
round(x_eigen$vectors,3)==round(my_svd$u,3)
##      [,1] [,2]
## [1,] TRUE TRUE
## [2,] TRUE TRUE

Problem Set 2

Describing the steps my function will take

To find the inverse of matrix A

\[A^{-1} = 1/det(A)*adj(A)\]

To find the adjunct of matrix A \[adj(A) =[C_{row,col}]^t \]

The cofactor matrix is the results of \[ [C_{row,col}]=-1^{row+col} * det (Minor_{row,col}) \]

Steps for function

  1. Determine if matrix is invertible(determinate!=0)

  2. look at each row and column of oriignal matrix and calculate the cofactor matrix by performing the operation above \[ [C_{row,col}]=-1^{row+col} * det (Minor_{row,col}) \]

  3. take the transpose of that matrix and multiply it by 1/determinate of original matrix

  4. This step is done outside of the matrix,

get_inverse <- function(matrix){
if (det(matrix)== 0) {
    print("matrix is not invertible")
    }
## get matrix size
rows <- dim(matrix)[1]
co_factor_matrix <- matrix(0,rows,ncol=rows)

## augment co_factor matrix by using adjugate formula
for (row in (1:rows)) {
    for (col in (1:rows)){
        co_factor_matrix[row,col] <- (-1)^(row+col)* det(matrix[-row,-col],ncol=rows-1)
    }
}
returned_matrix <- (1/det(matrix))*t(co_factor_matrix) 
return(returned_matrix)
}

Test function

mat_A <- matrix(c(1,0,1,2,4,0,3,5,6),ncol=3)

round(get_inverse(mat_A),3)==round(solve(mat_A),3)
##      [,1] [,2] [,3]
## [1,] TRUE TRUE TRUE
## [2,] TRUE TRUE TRUE
## [3,] TRUE TRUE TRUE
get_inverse(mat_A)%*%mat_A
##      [,1]          [,2]          [,3]
## [1,]    1 -8.881784e-16 -1.332268e-15
## [2,]    0  1.000000e+00  0.000000e+00
## [3,]    0  0.000000e+00  1.000000e+00
  • above example taken from link