1 Problem Set

1.1

\(A^TA\neq AA^T\)

Prof: For a non-squire matrix, \(n \times m\), \(A^TA\) produces \(m \times m\) matrix, and \(AA^T\) produces \(n \times n\) matrix.

See an example below.

a <- matrix(1:12, ncol = 3, nrow = 4)

at<-t(a)

aat<-a%*%at

aat
##      [,1] [,2] [,3] [,4]
## [1,]  107  122  137  152
## [2,]  122  140  158  176
## [3,]  137  158  179  200
## [4,]  152  176  200  224
ata<-at%*%a

ata
##      [,1] [,2] [,3]
## [1,]   30   70  110
## [2,]   70  174  278
## [3,]  110  278  446

For a squire matrix, the \(A^TA= AA^T\) only if a matrix is symmetrical ( \(a_{ij} =a_{ji}\)).

See, example below.

a <- matrix(c(1,2,3,2,4,5,3,5,6), ncol = 3, nrow = 3)

a
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    2    4    5
## [3,]    3    5    6
at<-t(a)

aat<-a%*%at

aat
##      [,1] [,2] [,3]
## [1,]   14   25   31
## [2,]   25   45   56
## [3,]   31   56   70
ata<-at%*%a

ata
##      [,1] [,2] [,3]
## [1,]   14   25   31
## [2,]   25   45   56
## [3,]   31   56   70

It a squire matrix is not symetrical, then equality won’t hold.

See example below.

a <- matrix(1:9, ncol = 3, nrow = 3)

at<-t(a)

aat<-a%*%at

aat
##      [,1] [,2] [,3]
## [1,]   66   78   90
## [2,]   78   93  108
## [3,]   90  108  126
ata<-at%*%a

ata
##      [,1] [,2] [,3]
## [1,]   14   32   50
## [2,]   32   77  122
## [3,]   50  122  194

2 Problem set

A<-matrix(c(1,7,6,2,9,5,3,7,5,89,3,67,67,45,3,5),ncol=4,nrow=4)

A
##      [,1] [,2] [,3] [,4]
## [1,]    1    9    5   67
## [2,]    7    5   89   45
## [3,]    6    3    3    3
## [4,]    2    7   67    5
LU<-function(C){
  D<-diag(nrow(C))
  for (i in 2:(nrow(C)))
    {for (j in i:(nrow(C)))
      {
      temp<-C[j,i-1]/C[i-1,i-1]
      C[j,]<-C[j,]-(C[j,i-1]/C[i-1,i-1])*C[i-1,]
      D[j,i-1]<-temp
      }
    }
    print(C)
    print(D)
    return(D%*%C)
            }

D<-LU(A)
##      [,1] [,2]      [,3]       [,4]
## [1,]    1    9   5.00000   67.00000
## [2,]    0  -58  54.00000 -424.00000
## [3,]    0    0 -74.48276  -26.17241
## [4,]    0    0   0.00000  -65.01667
##      [,1]      [,2]       [,3] [,4]
## [1,]    1 0.0000000  0.0000000    0
## [2,]    7 1.0000000  0.0000000    0
## [3,]    6 0.8793103  1.0000000    0
## [4,]    2 0.1896552 -0.6277778    1
D
##      [,1] [,2] [,3] [,4]
## [1,]    1    9    5   67
## [2,]    7    5   89   45
## [3,]    6    3    3    3
## [4,]    2    7   67    5