Find a basis for the eigenspace corresponding to each listed

prelimaf1 2021-11-21 Answered
Find a basis for the eigenspace corresponding to each listed eigenvalue of A below.
\[A=\begin{bmatrix}1&0\\-1&2\end{bmatrix},\lambda=2,1\]

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Expert Answer

Donald Proulx
Answered 2021-11-22 Author has 1830 answers

Given matrix A is
\[A=\begin{bmatrix}1&0\\-1&2\end{bmatrix}\]
f it;s eigen value \(\displaystyle\lambda={2},{1}\)
Next find eigen vector corresponding eigen value.
For \(\displaystyle\lambda={2}\)
\(\displaystyle\Rightarrow{A}{x}=\lambda{x}\)
\[\Rightarrow\begin{bmatrix}1&0\\-1&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=2\begin{bmatrix}x_1\\x_2\end{bmatrix}\]
\(\displaystyle\Rightarrow{x}_{{2}}={2}{x}_{{1}}\)
\(\displaystyle-{x}_{{1}}+{2}{x}_{{2}}={2}{x}_{{2}}\)
\(\displaystyle\Rightarrow{x}_{{1}}-{2}{x}_{{1}}={0}\)
\(\displaystyle\Rightarrow-{x}_{{1}}={0}\)
\(\displaystyle\Rightarrow{x}_{{2}}={1}\)
\[\Rightarrow x=\begin{bmatrix}0\\1\end{bmatrix}\]
For \(\displaystyle\lambda={1}\)
\(\displaystyle\Rightarrow{A}{x}=\lambda x\)
[\Rightarrow\begin{bmatrix}1&0\\-1&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}x_1\\x_2\end{bmatrix}\]
\(\displaystyle\Rightarrow{x}_{{1}}={x}_{{1}}\)
\(\displaystyle-{x}_{{1}}+{2}{x}_{{2}}={x}_{{2}}\)
\(\displaystyle\Rightarrow{x}_{{1}}={x}_{{2}}\)
\[x=\begin{bmatrix}1\\1\end{bmatrix}\]
is a basis for the eigenspace corresponding to \(\displaystyle\lambda={1}\)

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