Q1

Find X

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

Find Y

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

Eigenvalues & Eigenvectors

## eigen() decomposition
## $values
## [1] 26.601802  4.398198
## 
## $vectors
##           [,1]       [,2]
## [1,] 0.6576043 -0.7533635
## [2,] 0.7533635  0.6576043
## eigen() decomposition
## $values
## [1] 2.660180e+01 4.398198e+00 1.058982e-16
## 
## $vectors
##             [,1]       [,2]       [,3]
## [1,] -0.01856629 -0.6727903  0.7396003
## [2,]  0.25499937 -0.7184510 -0.6471502
## [3,]  0.96676296  0.1765824  0.1849001

Left-Singular, Singular, Right-Singular Values

## [1] TRUE TRUE
## [1] TRUE TRUE TRUE

This comparison provies that the first columns of the eigenvectors and their respective singular columns are exactly the same except multiplied by -1.

Non-Singular Values

## [1] TRUE TRUE

This shows that both X and Y are the same and are the squares of the non-zero singular values of A

Q2

##             [,1]        [,2]       [,3]
## [1,] -0.02857143 -0.05714286  0.2857143
## [2,]  0.28571429 -0.42857143  0.1428571
## [3,]  0.11428571  0.22857143 -0.1428571

Inverse Function

Check

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