EE.C26

For matrix
\[A = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\ \end{bmatrix} \]

the characteristic polynomial of A is \(^PA(x)=(4-x)(1−x)^2\) Find the eigenvalues and corresponding eigenspaces of A.

Solution:

\(Av = λv\) \((λI^n - A) v = 0v\) \(det(λI^n -A) = 0\)

\[λI^3 - A = \begin{bmatrix} λ-2 & -1 & -1 \\ -1 & λ-2 & -1 \\ -1 & -1 & λ-2 \\ \end{bmatrix} \]

\[λI^3 - A = \begin{bmatrix} λ-2 & -1 & -1 \\ -1 & λ-2 & -1 \\ -1 & -1 & λ-2 \\ \end{bmatrix} * \begin{bmatrix} λ-2 & -1 \\ -1 & λ-2 \\ -1 & -1 \\ \end{bmatrix} \]

\(=(λ-2)(λ-2)λ-2) + (-1)(-1)(-1) + (-1)(-1)(-1) - (-1)(-1)(λ-2) - (λ-2)(-1)(-1) - (-1)(λ-2)(-1)\)

\(p((λ) = λ^3 - 6λ^2 +9λ - 4 = 0\)

Factoring the polynomial gives us \((λ-4)(λ-1)(λ-1)\)

Eigenvalues are λ=4 and λ=1.

Eigenvectors:

for

for λ = 4, \[ \begin{bmatrix} 2 & -1 & -1\\ -1 & 2 & -1 \\ -1 & -1 & 2 \\ \end{bmatrix} \]

RREF for λ = 4 is

\[ \begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \]

and the eigenspace is

\[ \begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix} \]

a = matrix(c(2,-1,-1,-1,2,-1,-1,-1,2), nrow = 3)
eigen(a)
## eigen() decomposition
## $values
## [1] 3.000000e+00 3.000000e+00 2.220446e-15
## 
## $vectors
##            [,1]       [,2]      [,3]
## [1,]  0.0000000  0.8164966 0.5773503
## [2,] -0.7071068 -0.4082483 0.5773503
## [3,]  0.7071068 -0.4082483 0.5773503

for λ = 1, \[ \begin{bmatrix} -1 & -1 & -1\\ -1 & -1 & -1 \\ -1 & -1 & -1 \\ \end{bmatrix} \] RREF for λ = 1 is

\[ \begin{bmatrix} -1 &-1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \]

and the eigenspace is

\[ \begin{bmatrix} 1 \\ -1 \\ 0 \\ \end{bmatrix} \] \[ \begin{bmatrix} 1 \\ 0 \\ -1 \\ \end{bmatrix} \]

a = matrix(c(-1,-1,-1,-1,1,-1,-1,-1,1), nrow = 3)
eigen(a)
## eigen() decomposition
## $values
## [1]  2  1 -2
## 
## $vectors
##            [,1]       [,2]      [,3]
## [1,]  0.0000000  0.5773503 0.8164966
## [2,] -0.7071068 -0.5773503 0.4082483
## [3,]  0.7071068 -0.5773503 0.4082483