Find the eigenvalues, eigenspaces, algebraic multiplicities and geometric multiplicities for the matrix below. It is possible to do all these computations by hand, and it would be instructive to do so.
\[ C=\begin{bmatrix} -1 & 2 \\ -6 & 6 \\ \end{bmatrix} \]
Finding the eigenvalues \(\lambda\): \[det(A- \lambda I)=0\] \[ det(\begin{bmatrix} a- \lambda & b \\ c & d- \lambda \\ \end{bmatrix}) =0 \] \[(a- \lambda)(d- \lambda)-bc=0\] \[(-1- \lambda)(6- \lambda)-(2)(-6)=0\] \[-6-6 \lambda+ \lambda + \lambda^2+12=0\] \[ \lambda^2-5 \lambda+6=0\] \[ ( \lambda-2)( \lambda-3)=0\] \[ \lambda = 2, 3\]
Finding the eigenspaces:
For \(\lambda = 2\)
\(A- \lambda I\)
\[ \begin{bmatrix} -1 & 2 \\ -6 & 6 \\ \end{bmatrix} - \begin{bmatrix} 2 & 0 \\ 0 & 2 \\ \end{bmatrix} \] \[ \begin{bmatrix} -3 & 2 \\ -6 & 4 \\ \end{bmatrix} \] \[ \begin{bmatrix} -3 & 2 \\ -6 & 4 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix} =0 \]
\[ -3x+2y=0 \]
\[ -6x+4y=0 \]
\[ -3x+2y=0 \]
\[ -3x=-2y \]
\[ v = \begin{bmatrix} 2 \\ 3 \\ \end{bmatrix} \]
The eigenspace associated with the eigenvalue 2 is the span of the vector \(\begin{bmatrix} 2 \\ 3 \\ \end{bmatrix}\).
For \(\lambda = 3\)
\(A- \lambda I\)
\[ \begin{bmatrix} -1 & 2 \\ -6 & 6 \\ \end{bmatrix} - \begin{bmatrix} 3 & 0 \\ 0 & 3 \\ \end{bmatrix} \]
\[ \begin{bmatrix} -4 & 2 \\ -6 & 3 \\ \end{bmatrix} \]
\[ \begin{bmatrix} -4 & 2 \\ -6 & 3 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix} =0 \]
\[ -4x+2y=0 \]
\[ -6x+3y=0 \]
\[ -4x+2y=0 \]
\[ -4x=-2y \]
\[ v = \begin{bmatrix} 1 \\ 2 \\ \end{bmatrix} \]
The eigenspace associated with the eigenvalue 3 is the span of the vector \(\begin{bmatrix} 1 \\ 2 \\ \end{bmatrix}\).
Finding algebraic and geometric multiplicities: Since both 2 and 3 are distinct roots of the quadratic equation we solved, both have algebraic multiplicities of 1. Since we found one linearly independent vector associated with each eigenvalue, both have geometric multiplicities of 1.
Check with R functions:
library(Matrix)
A <- matrix(c(-1, 2, -6, 6), nrow = 2, byrow = TRUE)
eigen(A)## eigen() decomposition
## $values
## [1] 3 2
##
## $vectors
## [,1] [,2]
## [1,] -0.4472136 -0.5547002
## [2,] -0.8944272 -0.8320503