DATA605 Wk3 Discussion3

For matrix

\[\mathbf{A} = \left[\begin{array} {rrr} 0 & 4 & -1 & 1\\ -2 & 6 & -1 & 1\\ -2 & 8 & -1 & -1\\ -2 & 8 & -3 & 1 \end{array}\right]\]

the characteristics polynomial of \(A\) is \(p_A(x) = (x +2)(x - 2)^2(x-4)\). Find the eigenvalues and corresponding eigenspaces of A.

library(matlib)
library(pracma)

## 
## Attaching package: 'pracma'

## The following objects are masked from 'package:matlib':
## 
##     angle, inv

A <- matrix(c(0, 4, -1, 1, -2, 6, -1, 1, -2, 8, -1, -1, -2, 8, -3, 1), 4, 4, byrow = T)
I <- diag(4)

\(\lambda = -2\)

\(A+2I\)

A + (2*I)

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

#Reduced Row Echelon Form
rref((A + (2*I)))

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

Eigenvector for \(\lambda = -2\):

\[\left[\begin{array} {rrr} 0\\0\\1\\1 \end{array}\right]\]

\(\lambda = 2\)

\(A-2I_4\)

A - (2*I)

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

#Reduced Row Echelon Form
rref((A - (2*I)))

##      [,1] [,2] [,3] [,4]
## [1,]    1    0 -0.5 -1.5
## [2,]    0    1 -0.5 -0.5
## [3,]    0    0  0.0  0.0
## [4,]    0    0  0.0  0.0

Eigenvectors for \(\lambda = 2\):

\[\left[\begin{array} {rrr} 1\\1\\2\\0 \end{array}\right] \left[\begin{array} {rrr} 3\\1\\0\\2 \end{array}\right]\]

---

\(\lambda = 4\)

\(A-4I_4\)

A - (4*I)

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

#Reduced Row Echelon Form
rref((A - (4*I)))

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

Eigenvector for \(\lambda = 4\):

\[\left[\begin{array} {rrr} 1\\1\\1\\1 \end{array}\right]\]

---

Corresponding eigenspaces of A.

\(\gamma A\) (-2) = 1, \(\gamma A\) (2) = 2 and \(\gamma A\) (4) = 1