Chapter E Section EE Exercise C24

Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for

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

Solution

A <- matrix(c(1,-1,1,-1,1,-1,1,-1,1), nrow=3, byrow=TRUE)
A
##      [,1] [,2] [,3]
## [1,]    1   -1    1
## [2,]   -1    1   -1
## [3,]    1   -1    1

Eigenvalues

# Calculate characteristic polynomial using pracma library
library(pracma)
charpoly(A)
## [1]  1 -3  0  0

\(p_A(x) = x^3-3x^2+0x+0 = x^2(x-3)\)

\(p_A(\lambda) = 0\), so eigenvalues are \(\lambda = 0\) and \(\lambda = 3\).

Algebraic multiplicities

\(\alpha_A(0) = 2\) and \(\alpha_A(3) = 1\)

Eigenspace

If \(\lambda=0\), then \(A - 0I_3 = A\) and it is row-reduced to

rref(A)
##      [,1] [,2] [,3]
## [1,]    1   -1    1
## [2,]    0    0    0
## [3,]    0    0    0

Then eigenspace is
\[ \Bigg\langle \Bigg\{ \begin{bmatrix} 1\\ 1\\ 0 \end{bmatrix} , \begin{bmatrix} -1\\ 0\\ 1 \end{bmatrix} \Bigg\} \Bigg \rangle \]

If \(\lambda=3\), then \(A - 3I_3\) is row-reduced to

rref(A - 3 * diag(3))
##      [,1] [,2] [,3]
## [1,]    1    0   -1
## [2,]    0    1    1
## [3,]    0    0    0

Then eigenspace is
\[ \Bigg\langle \Bigg\{ \begin{bmatrix} 1\\ -1\\ 1 \end{bmatrix} \Bigg\} \Bigg \rangle \]

Geometric multiplicities

\(\gamma_A(0) = 2\) and \(\gamma_A(3) = 1\)

Notes

eigen function again returns a very small, near-zero value instead of \(0\).

eigen(A)$values
## [1] 3.000000e+00 8.881784e-16 0.000000e+00

Additionally, per theorem DMFE since \(\alpha_A(\lambda) = \gamma_A(\lambda)\) for all \(\lambda\), then \(A\) should be diagonalizable and eigenvalues should be on the diagonal. Consider \(S\) consisting of eigenvectors.

S <- matrix(c(1,-1,1,1,0,-1,0,1,1), nrow=3, byrow=TRUE)
S
##      [,1] [,2] [,3]
## [1,]    1   -1    1
## [2,]    1    0   -1
## [3,]    0    1    1

Compute \(S^{-1}AS\)

inv(S) %*% A %*% S
##      [,1] [,2]          [,3]
## [1,]    0    0 -1.665335e-16
## [2,]    0    0 -3.330669e-16
## [3,]    0    0  3.000000e+00

Not considering near-zero values, everything checks out.