Discussion 3

C21

Comment to C21 posted by Yuen Chun Way

#create matrix A
A = array( 
   c(-1,-6,2,6),  
   dim=c(2,2))

print("matrix A")

## [1] "matrix A"

print(A)

##      [,1] [,2]
## [1,]   -1    2
## [2,]   -6    6

#create matrix A
C = array( 
   c(1,0,0,1),  
   dim=c(2,2))

#eigenvalues
print("eigenvalues")

## [1] "eigenvalues"

eigenvalues <- eigen(A)
round(eigenvalues$values,0)

## [1] 3 2

eigen_space <- function(eig_v){
  
print("matrix I")
   eig_v_I = array( 
   c(eig_v,0,0,eig_v),  
   dim=c(2,2))

B <- A-eig_v_I
print("matrix A-eigenvalue*I")
print(B)

#B[1,1]x1=-B[2,2]x2
#B[2,1]x1=-B[1,2]x2

print(B[2,1])
x1 <- -B[1,1]/B[1,2]
x2 <- -B[2,2]/B[2,1]

print("x1")
print(x1)
print("x2")
print(x2)

}

eigen_space(3)

## [1] "matrix I"
## [1] "matrix A-eigenvalue*I"
##      [,1] [,2]
## [1,]   -4    2
## [2,]   -6    3
## [1] -6
## [1] "x1"
## [1] 2
## [1] "x2"
## [1] 0.5

eigen_space(2)

## [1] "matrix I"
## [1] "matrix A-eigenvalue*I"
##      [,1] [,2]
## [1,]   -3    2
## [2,]   -6    4
## [1] -6
## [1] "x1"
## [1] 1.5
## [1] "x2"
## [1] 0.6666667

C27

For matrix

\[A = \begin{bmatrix} 0 & 4 & -1 & 1 \\ -2 & 6 & -1 & 1 \\ -2 & 8 & -1 & -1 \\ -2 & 8 & -3 & 1 \\ \end{bmatrix}\]

, the characteristic polynomial of A is pA(x) =\((x + 2)\)\((x − 2)^2\)\((x − 4)\) Find the eigenvalues and corresponding eigenspaces of A.

#create matrix A
A = array( 
   c(0,-2,-2,-2,4,6,8,8,-1,-1,-1,-3,1,1,-1,1),  
   dim=c(4,4))

print(A)

##      [,1] [,2] [,3] [,4]
## [1,]    0    4   -1    1
## [2,]   -2    6   -1    1
## [3,]   -2    8   -1   -1
## [4,]   -2    8   -3    1

#calculate eigenvalues
eigenvalues <- eigen(A)
round(eigenvalues$values,0)

## [1]  4 -2  2  2

#build function for calculating eigenspaces

eigenspace <- function(eig_v) {
  
eig_v_I = array( 
   c(eig_v,0,0,0,0,eig_v,0,0,0,0,eig_v,0,0,0,0,eig_v),  
   dim=c(4,4))
print(eig_v_I)

#printeigenspace
print(A-eig_v_I)
}

#eigenspaces for corresponding eigenvalues
eigenspace(2)

##      [,1] [,2] [,3] [,4]
## [1,]    2    0    0    0
## [2,]    0    2    0    0
## [3,]    0    0    2    0
## [4,]    0    0    0    2
##      [,1] [,2] [,3] [,4]
## [1,]   -2    4   -1    1
## [2,]   -2    4   -1    1
## [3,]   -2    8   -3   -1
## [4,]   -2    8   -3   -1

eigenspace(-2)

##      [,1] [,2] [,3] [,4]
## [1,]   -2    0    0    0
## [2,]    0   -2    0    0
## [3,]    0    0   -2    0
## [4,]    0    0    0   -2
##      [,1] [,2] [,3] [,4]
## [1,]    2    4   -1    1
## [2,]   -2    8   -1    1
## [3,]   -2    8    1   -1
## [4,]   -2    8   -3    3

eigenspace(4)

##      [,1] [,2] [,3] [,4]
## [1,]    4    0    0    0
## [2,]    0    4    0    0
## [3,]    0    0    4    0
## [4,]    0    0    0    4
##      [,1] [,2] [,3] [,4]
## [1,]   -4    4   -1    1
## [2,]   -2    2   -1    1
## [3,]   -2    8   -5   -1
## [4,]   -2    8   -3   -3

#calculate eigenvectors
round(eigenvalues$vectors,0)

##      [,1] [,2] [,3] [,4]
## [1,]   -1    0   -1   -1
## [2,]   -1    0    0    0
## [3,]   -1   -1   -1    0
## [4,]    0   -1    0    0