C24 pg. 387
Find the eigenvalues, eigenspaces, algebraic and geometric multiplities for
\[\mathbf{A} = \begin{bmatrix} 1 & -1 & 1 \\ -1 & 1 & -1 \\ 1 & -1 & 1 \end{bmatrix} \]
# the matrix
A = matrix(c(1,-1,1,-1,1,-1,1,-1,1), nrow = 3, byrow = T)In order to compute \(\lambda\) value, we have
\(det(A - \lambda I) = 0\)
\(det(\begin{bmatrix} 1&-1&1\\-1&1&-1\\1&-1&1\end{bmatrix} - \begin{bmatrix} \lambda&0&0\\0&\lambda&0\\0&0&\lambda\end{bmatrix}) = 0\)
\(det(\begin{bmatrix} 1-\lambda&-1&1\\-1&1-\lambda&-1\\1&-1&1-\lambda\end{bmatrix}) = 0\)
\(=(1-\lambda)\begin{bmatrix} 1-\lambda&-1\\-1&1-\lambda\end{bmatrix}-(-1)\begin{bmatrix} -1&-1\\1&1-\lambda\end{bmatrix}+1\begin{bmatrix} -1&1-\lambda\\1&-1\end{bmatrix}\)
\(=(1-\lambda)[(1-\lambda)^2-1]+\lambda-1+1+1-(1-\lambda)\)
\(=(1-\lambda)(1-\lambda-1)(1-\lambda+1)+2\lambda\)
\(=(1-\lambda)(-\lambda)(2-\lambda)+2\lambda\)
\(=(\lambda^2-\lambda)(2-\lambda)+2\lambda\)
\(=2\lambda^2-\lambda^3-2\lambda+\lambda^2+2\lambda\)
\(=-\lambda^3+3\lambda^2\)
\(=\lambda^2(-\lambda+3)\)
Therefore, \(\lambda \in \{0,3\}\), the eigenvalues are 0 and 3.
The algebraic multiplicities(the power of corresponding \((x-\lambda)\)) \(\alpha_A(3) = 1\) and \(\alpha_A(0) = 2\)
library(pracma)
I = matrix(c(1,0,0,0,1,0,0,0,1), nrow = 3, byrow = T)
# when lambda = 3, we have
detA = A - 3 * I
# row reduce
rref(detA)## [,1] [,2] [,3]
## [1,] 1 0 -1
## [2,] 0 1 1
## [3,] 0 0 0the eigenspace for \(\lambda = 3\) is \[\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}\]
#similarly, when lambda = 0, we have
detA1 = A - 0 * I
rref(detA1)## [,1] [,2] [,3]
## [1,] 1 -1 1
## [2,] 0 0 0
## [3,] 0 0 0the eigenspace for \(\lambda = 0\) is \[\begin{bmatrix} 1\\1\\0\end{bmatrix}, \begin{bmatrix} -1\\0\\1\end{bmatrix}\]
The geometric multiplicities(the dimension of the vector) are \(\gamma_A(3) = 1\) and \(\gamma_A(0) = 1\)