—————————————————————————

Student Name : Sachid Deshmukh

—————————————————————————

Problem set 1

1. Matrix A = 

A <- matrix(c(1,-1,2,0,3,4), nrow=2)
print(A)
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]   -1    0    4

2. Matrix X = 

X <- A %*% t(A)
print(X)
##      [,1] [,2]
## [1,]   14   11
## [2,]   11   17

3. Matrix Y = 

Y <-   t(A) %*% A
print(Y)
##      [,1] [,2] [,3]
## [1,]    2    2   -1
## [2,]    2    4    6
## [3,]   -1    6   25

4. Sungular Value Decomposition of A

eigenX <- eigen(X)
eigenY <- eigen(Y)
svdA <- svd(A)
print(svdA)
## $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

5. Validate that U is same as Eigen Vector of X 

eigenX$vectors[,1] <- eigenX$vectors[,1] * (-1)
round(eigenX$vectors, digits=10) == round(svdA$u, digits=10)
##      [,1] [,2]
## [1,] TRUE TRUE
## [2,] TRUE TRUE

6. Validate that V is same as Eigen Vector of Y 

eigenY$vectors[,1] <- eigenY$vectors[,1] * (-1)
round(eigenY$vectors[,1:2], digits=10) == round(svdA$v, digits=10)
##      [,1] [,2]
## [1,] TRUE TRUE
## [2,] TRUE TRUE
## [3,] TRUE TRUE

7. Validate that sigma is same as Square Root of Eigen Values of X 

round(eigenX$values, digits=10) == round(svdA$d^2, digits=10)
## [1] TRUE TRUE

Problem set 2

1. myinverse function definition

myinverse <- function(X) {
  
  # Get Matrix Dimension
  n <- dim(X)[1]
  
  # Matrix Inversion Loop
  Y <- matrix(0, n, n)
  for(i in 1:n) {
    for(j in 1:n) {
      Y[i,j] <- (-1)^(i+j)*det(matrix(X[-i,-j], n-1))
    }
  }
  
  # Calculate Inverse
  Z <- t(Y)/det(X)
  
  return(Z)
}

2. Test myinverse function

A <- matrix(c(5,3,7,2,8,9,3,5,7), nrow=3)
print(A)
##      [,1] [,2] [,3]
## [1,]    5    2    3
## [2,]    3    8    5
## [3,]    7    9    7
B <- myinverse(A)
print(B)
##       [,1]  [,2] [,3]
## [1,] -2.75 -3.25  3.5
## [2,] -3.50 -3.50  4.0
## [3,]  7.25  7.75 -8.5

3. Validate that A.B = Identity Matrix

I = round(A %*% B, digits=10)
print(I)
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1