C 19, p.389

Task

Find the eigenvalues, eigenspaces, algebraic multiplicities and geometric multiplici- ties for the matrix below. It is possible to do all these computations by hand, and it would be instructive to do so.

\[ A = \begin{bmatrix} -1 & 2 \\ -6 & 6 \end{bmatrix} \]

Eigenvalues

To find eigenvalues and eigenvectors, we take the determinant of a special matrix and set it to zero: det(A−λI) = 0

\[ det ( \begin{bmatrix} -1 & 2 \\ -6 & 6 \end{bmatrix} - \begin{bmatrix} λ & 0 \\ 0 & λ \end{bmatrix} ) = 0 \]

(-1 - λ) * (6 - λ) + 12 = 0

λ2 - 5λ + 6 = 0

(λ - 3) * (λ - 2) = 0

λ ∈ {3, 2}

Eigenspaces

Since we have two distinct eigenvalues, we will have two eigenvectors.

To find these eigenvectors, we must have that it satisfies the equation (A - λI)v = 0

  1. for λ = 3
##      [,1] [,2]
## [1,]   -4    2
## [2,]   -6    3

v = (v1, v2) = v1(2,1), so (2,1) is a basis for that eigenspace with eigenvalue λ = 3.

  1. for λ = 2
##      [,1] [,2]
## [1,]   -3    2
## [2,]   -6    4

v = (v1, v2) = v1(3,2), so (3,2) is a basis for that eigenspace with eigenvalue λ = 2.

Algebraic and Geometric Multiplicity

Algebraic and Geometric Multiplicity equals 1 for both eigenvalues.

Checking eigenvalues with R

## eigen() decomposition
## $values
## [1] 3 2
## 
## $vectors
##            [,1]       [,2]
## [1,] -0.4472136 -0.5547002
## [2,] -0.8944272 -0.8320503