finding eigenvectors

**Jskid** · March 18th 2011, 09:07 PM

Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $ \[ A = \begin{array}{ccc} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \\ \end{array} \]$
I got the eigenvalues $\lambda_1=\lambda_2=\lambda_3=0$
To find the associated eigenvectors $(0I_3-A)\vec x = \vec 0$
This gives me the matrix $ \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

**dwsmith** · March 18th 2011, 09:17 PM

Originally Posted by Jskid

Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $ \[ A = \begin{array}{ccc} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \\ \end{array} \]$
I got the eigenvalues $\lambda_1=\lambda_2=\lambda_3=0$
To find the associated eigenvectors $(0I_3-A)\vec x = \vec 0$
This gives me the matrix $ \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

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

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

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