C24

C24 Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for

\[\mathbf{A} = \left[\begin{array}{rrr}1 & -1 & 1\\-1 & 1 & -1\\1 & -1 & 1\end{array}\right] \]

A <- matrix(c(1,-1,1, -1,1,-1, 1,-1,1), nrow=3, ncol=3, byrow=TRUE)
paste0("A=")

## [1] "A="

A

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

(a) Find eigenvalues

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"

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"

(b) Find eigenspaces

I3 <- matrix(c(1,0,0, 0,1,0, 0,0,1), nrow=3, ncol=3, byrow=TRUE)

print("eigenspace for eigenvalue = 3")

## [1] "eigenspace for eigenvalue = 3"

(NS_3 <- nullspace(A-(3*I3)))

##            [,1]
## [1,]  0.5773503
## [2,] -0.5773503
## [3,]  0.5773503

print("eigenspace for eigenvalue = 0")

## [1] "eigenspace for eigenvalue = 0"

(NS_0 <- nullspace(A-(0*I3)))

##           [,1]       [,2]
## [1,] 0.5773503 -0.5773503
## [2,] 0.7886751  0.2113249
## [3,] 0.2113249  0.7886751

print("eigenspace for eigenvalue = 3")

## [1] "eigenspace for eigenvalue = 3"

(reduce_3 <- rref(A-(3*I3)))

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

print("eigenspace for eigenvalue = 0")

## [1] "eigenspace for eigenvalue = 0"

(reduce_0 <- rref(A-(0*I3)))

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

print("eigenspace for eigenvalue = 3")

## [1] "eigenspace for eigenvalue = 3"

(NS_three <- matrix(c(1,-1,1), nrow=3, ncol=1, byrow=TRUE))

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

print("eigenspace for eigenvalue = 0")

## [1] "eigenspace for eigenvalue = 0"

(NS_zero <- matrix(c(1,1,0, -1,0,1), nrow=3, ncol=2, byrow=TRUE))

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

(c) Find algebraic multiplicities

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

Algebraic multiplicities : \[ \alpha_A(3)=1 \] and \[ \alpha_A(0)=2 \]

(d) Find geometric multiplicities

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