Exercise C20 Pg. 306

Find the eigenvalues, eigenspace, algebreic mulitplicities and geometric multiplicities for the matrix below

\[\begin{bmatrix} -12 & 30\\ -5 & 13 \end{bmatrix}\]

1. Finding the eighenvalues and algebraic multiplicities

\((-12 - \lambda)(13 - \lambda) - (30 * 5)\)

\(-156 - 13\lambda + 12\lambda +\) \(\lambda^2\)

\((\lambda-3)(\lambda+2)\)

eigenvalues \(\lambda\) = 3, −2

algebraic multiplicities (3) = 1 and (−2) = 1.

2. Finding the geometric multiplicities

For \(\lambda\) = 3

Find the reduced row echelon form of \(A - 3I_{2}\)

A = matrix(c(-12,30,-5,13), nrow = 2, byrow=T)
I = matrix(c(1,0,0,1), nrow = 2, byrow=T)

A3 <- A - (3 *  I)

A3[2,] <- (A3[2,] * -3) + A3[1,]
A3[1,] <- A3[1,] / -15

A3
##      [,1] [,2]
## [1,]    1   -2
## [2,]    0    0

\(E_{3}\) =

\[\begin{bmatrix} 2\\ 1 \end{bmatrix}\]

For \(\lambda\) = -2

Find the reduced row echelon form of \(A + 2I_{2}\)

A_2 <- A - (-2 *  I)
A_2
##      [,1] [,2]
## [1,]  -10   30
## [2,]   -5   15
A_2[2,] <- (A_2[2,] * - 2) + A_2[1,]
A_2[1,] <- A_2[1,] / -10

A_2
##      [,1] [,2]
## [1,]    1   -3
## [2,]    0    0

\(E_{-2}\) =

\[\begin{bmatrix} 3\\ 1 \end{bmatrix}\]