Data 605-Week 3 Discussion Question C24 - p.388

library(pracma)

Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for A =

A <- matrix(c(1,-1,1,-1,1,-1,1,-1,1),nrow=3)
A

##      [,1] [,2] [,3]
## [1,]    1   -1    1
## [2,]   -1    1   -1
## [3,]    1   -1    1

eigenvalues <- eigen(A)
round(eigenvalues$values, digits=14)

## [1] 3 0 0

eigenvalues$vectors

##            [,1]       [,2]      [,3]
## [1,]  0.5773503 -0.8164966 0.0000000
## [2,] -0.5773503 -0.4082483 0.7071068
## [3,]  0.5773503  0.4082483 0.7071068

charpoly(A)

## [1]  1 -3  0  0

geometric_multiplicities <- A - 3*diag(3)
Result<- rref(geometric_multiplicities)
Result

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