# finding eigenvectors

• Mar 18th 2011, 09:07 PM
Jskid
finding eigenvectors
Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $\displaystyle $A = \begin{array}{ccc} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \\ \end{array}$$
I got the eigenvalues $\displaystyle \lambda_1=\lambda_2=\lambda_3=0$
To find the associated eigenvectors $\displaystyle (0I_3-A)\vec x = \vec 0$
This gives me the matrix $\displaystyle \begin{array}{cccc} 0 & -1 & -2 & 0 \\ 0 & 0 & -3 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}$ where the last column of 0s is the augmented part of the matrix.
This matrix is row equivalent to the identity matrix and that confuses me
• Mar 18th 2011, 09:17 PM
dwsmith
Quote:

Originally Posted by Jskid
Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $\displaystyle $A = \begin{array}{ccc} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \\ \end{array}$$
I got the eigenvalues $\displaystyle \lambda_1=\lambda_2=\lambda_3=0$
To find the associated eigenvectors $\displaystyle (0I_3-A)\vec x = \vec 0$
This gives me the matrix $\displaystyle \begin{array}{cccc} 0 & -1 & -2 & 0 \\ 0 & 0 & -3 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}$ where the last column of 0s is the augmented part of the matrix.
This matrix is row equivalent to the identity matrix and that confuses me

$\displaystyle \displaystyle\begin{bmatrix}0&1&2\\0&0&3\\0&0&0\en d{bmatrix}\Rightarrow\text{rref}=\begin{bmatrix}0& 1&0\\0&0&1\\0&0&0\end{bmatrix}$

$\displaystyle \displaystyle\left\{\begin{bmatrix}1\\0\\0\end{bma trix}\right\}$