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C24. Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for \[\begin{equation*} A= \begin{bmatrix} 1&-1&1\\ -1&1&-1\\ 1&-1&1 \end{bmatrix} \end{equation*}\]

eigenvalues

(A <- matrix(c(1,-1,1,-1,1,-1,1,-1,1),3, byrow=T))
##      [,1] [,2] [,3]
## [1,]    1   -1    1
## [2,]   -1    1   -1
## [3,]    1   -1    1
eigen_a <- eigen(A)
eigenvalue <- eigen_a$values
print(paste("eigenvalues includes:",round(eigenvalue)))
## [1] "eigenvalues includes: 3" "eigenvalues includes: 0"
## [3] "eigenvalues includes: 0"

or eigenvalues could be calculated as follows:

suppressWarnings(suppressMessages(library(pracma)))
polyn <- charpoly(A,info=T) # find the characteristic polynomial
## Error term: 0
ev <- roots(polyn$cp) # eigenvalues are the roots of characteristic polynomials
print(paste("eigenvalues includes:",ev))
## [1] "eigenvalues includes: 0" "eigenvalues includes: 0"
## [3] "eigenvalues includes: 3"

algebraic multiplicities

Characteristic Polynomials is : \(-(x-3)x^2\)

algebraic multiplicities \(\alpha_A (3) = 1\) and \(\alpha_A (0) = 2\) .

eigenspaces

I3 <- matrix(c(1,0,0,0,1,0,0,0,1),3, byrow=T)
(NS_3 <- nullspace(A-3*I3)) # eigenspace for # eigenvalue = 3
##            [,1]
## [1,]  0.5773503
## [2,] -0.5773503
## [3,]  0.5773503
(NS_0 <-nullspace(A-0*I3)) # eigenspace for # eigenvalue = 0
##           [,1]       [,2]
## [1,] 0.5773503 -0.5773503
## [2,] 0.7886751  0.2113249
## [3,] 0.2113249  0.7886751
# or eigenspace could be calculated as follows:
(reduce_3 <- rref(A-3*I3))
##      [,1] [,2] [,3]
## [1,]    1    0   -1
## [2,]    0    1    1
## [3,]    0    0    0
(reduce_0 <- rref(A-0*I3))
##      [,1] [,2] [,3]
## [1,]    1   -1    1
## [2,]    0    0    0
## [3,]    0    0    0
(NS_three <- matrix(c(1,-1,1),3,byrow=T)) # eigenspace
##      [,1]
## [1,]    1
## [2,]   -1
## [3,]    1
(NS_zero <- matrix(c(1,1,0,-1,0,1),3,byrow=T)) #eigenspace
##      [,1] [,2]
## [1,]    1    1
## [2,]    0   -1
## [3,]    0    1

geometric multiplicities

So the eigenspace dimensions yield geometric multiplicities: \(\gamma_A (3) = 1\), \(\gamma_C (0) = 2\)