Discussion week3

Linear algebra text, Chapter E

Fundamentals of Computational Mathematics

CUNY MSDS DATA 605, Fall 2018

Rose Koh

09/11/2018

Chapter E

• Eigenvalues

We first compute the characteristic polynomial:

\[\mathbf{pc(x)} = det(C = xI_2)\] \[= \left[\begin{array} {rrr} -1-x & 2 \\ -6 & 6-x \\ \end{array}\right] \]

\[= (-1 -x)(6-x)-(2)(-6) \]

\[= x^2 - 5x + 6 \]

\[= (x-3)(x-2) \]

The eigenvalues of C are the solutions to pc(x) = 0, λ = 2 and λ = 3.

The eigenvalue only appears once in characteristic polynomial thus αC (2) = 1 and αC (3) = 1.

• Eigenspaces

In order to find the eigenspaces, we need to contruct the singular matrices and find expressions for the null spaces of these matrices.

\[ λ = 2 \] \[ C - (2)I_2 \] \[= \left[\begin{array} {rrr} -3 & 2 \\ -6 & 4 \\ \end{array}\right] \]

C2 <- matrix(c(-3,2,-6,4), 2, byrow=T)
library(pracma)
rref(C2)

##      [,1]       [,2]
## [1,]    1 -0.6666667
## [2,]    0  0.0000000

\[ E_c(2)=N(C−(2)I_2) \]

\[ x - 0.66y = 0 \] \[ x=\frac { 2 }{ 3 } y \]

\[ λ = 3 \] \[ C - (3)I_2 \] \[= \left[\begin{array} {rrr} -4 & 2 \\ -6 & 3 \\ \end{array}\right] \]

C3 <- matrix(c(-4,2,-6,3), 2, byrow=T)
rref(C3)

##      [,1] [,2]
## [1,]    1 -0.5
## [2,]    0  0.0

\[ E_c(3)=N(C−(3)I_2) \]

\[ x - 0.5y = 0 \]

\[ x=\frac { 1 }{ 2 } y \]

Each eigenspace has a single basis vector which means the dimensions are both 1 hence the geometric multiplicities are \[ γA (2) = 1 \] \[ γA (3) = 1\]

# check with eigen function
C <- matrix(c(-1,2,-6,6),2, byrow=T)
C

##      [,1] [,2]
## [1,]   -1    2
## [2,]   -6    6

eigen(C)$values

## [1] 3 2