DATA 605 Week03 Discussion

library(pracma)

## Warning: package 'pracma' was built under R version 3.3.3

Eigenvalues - Eigenvalues and Eigenvectors (E).C11 - Page 388

Manual Solution

\[\mathbf{A}=\left[\begin{matrix} 3 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \end{matrix}\right]; \mathbf{I}=\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]\]

\[\mathbf{p({\lambda})} = \mathbf{det(A + (-{\lambda}.I_3))}\]

\[\mathbf{p({\lambda})} = \mathbf{det(\left[\begin{matrix} 3 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \end{matrix}\right] + (-{\lambda} \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]))}\]

\[\mathbf{p({\lambda})} = \mathbf{det(\left[\begin{matrix} 3 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \end{matrix}\right] + \left[\begin{matrix} -{\lambda} & 0 & 0 \\ 0 & -{\lambda} & 0 \\ 0 & 0 & -{\lambda} \end{matrix}\right])}\]

\[\mathbf{p({\lambda})} = \mathbf{det(\left[\begin{matrix} 3-{\lambda} & 2 & 1 \\ 0 & 1-{\lambda} & 1 \\ 1 & 2 & -{\lambda} \end{matrix}\right])}\]

\[\mathbf{p({\lambda})} = \mathbf{det(\left[\begin{matrix} 3-{\lambda} & 2 & 1 \\ 0 & 1-{\lambda} & 1 \\ 1 & 2 & -{\lambda} \end{matrix}\right])} = (-{\lambda}+3).\left|\begin{matrix} -{\lambda}+1 & 1 \\ 2 & -{\lambda} \end{matrix}\right| + 0.\left|\begin{matrix} 2 & 1 \\ 2 & -{\lambda} \end{matrix}\right| + 1.\left|\begin{matrix} 2 & 1 \\ -{\lambda}+1 & 1 \end{matrix}\right| \]

\[\mathbf{p({\lambda})} = -{\lambda}^3+4{\lambda}^2-{\lambda}-6 + 0.\left|\begin{matrix} 2 & 1 \\ 2 & -{\lambda} \end{matrix}\right| + 1.\left|\begin{matrix} 2 & 1 \\ -{\lambda}+1 & 1 \end{matrix}\right| \]

\[\mathbf{p({\lambda})} = -{\lambda}^3+4{\lambda}^2-{\lambda}-6 + 0 + {\lambda}+1 \]

\[\mathbf{p({\lambda})} = -{\lambda}^3+4{\lambda}^2-5\]

\[\mathbf{p({\lambda})} = -(1.{\lambda}^3-4.{\lambda}^2+0.{\lambda}+5)\]

R Language Solution

A <- matrix(c(3, 2, 1, 0, 1, 1, 1, 2, 0), nrow = 3, ncol = 3, byrow = TRUE)
charpoly(A)

## [1]  1 -4  0  5