DATA 605 Week 3 Discussion

C11 Pg 306

Find the characteristic polynomial of

A <- matrix(c(3,2,1,0,1,1,1,2,0), nrow=3, byrow=T)
A

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

• To get the eigenvalues, we need to solve the determinant of this matrix;

\[\begin{bmatrix} 3 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \end{bmatrix}\]

• and subtract the determinant of following matrix from it, to get 0. This matrix below is just λ * \(I_n\), where \(I_n\) is the identity matrix for a 3x3 matrix;

\[\begin{bmatrix} λ & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & λ \end{bmatrix}\]

• That is, the determinant of the following matrix below, det(\(A - λI_n\)) = 0

\[\begin{bmatrix} 3-λ & 2 & 1 \\ 0 & 1-λ & 1 \\ 1 & 2 & 0-λ \end{bmatrix}\]

The characteristic polynomial is \(p(λ) = -λ^{3} + 4λ^{2} - 3λ - 4 = 0\)