Eigenvalues and eigenvectors: Eigenvalues and eigenvectors
Computing eigenvectors for a given eigenvalue
We start with examples to compute the eigenspace of an eigenvalue of a matrix.
Let \(\lambda = -1\) be an eigenvalue of the matrix \[A=\matrix{-2 & 0 \\ 2 & -1}\] is. Then there must be a vector \(\vec{v}\) such that \(A\vec{v}=-1\vec{v}\), that is, \[(A+I)\vec{v}=\vec{0}\text.\] In other words, we must find the kernel of the matrix \(A+I\).
We can do this through row reduction of the matrix \[A+I = \matrix{-2 & 0 \\ 2 & -1} - \matrix{-1 & 0 \\0& -1 }=\matrix{ -1 & 0 \\ 2 & 0}\] This can be done as follows:
\[\begin{aligned}
\matrix{-1&0\\2&0\\}&\sim\matrix{1&0\\2&0\\}&{\blue{\begin{array}{c}-R_1\\\phantom{x}\end{array}}}\\\\ &\sim\matrix{1&0\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2-2R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = -1\) equals \(\left\{ r \cv{0\\1} \middle|\;r\in\mathbb R\right\}\).
If needed, we avoided here fractions in the solution.
In other words, the eigenspace of eigenvalue \(-1\) equals \(\left\langle\cv{0\\1}\right\rangle\)
We can do this through row reduction of the matrix \[A+I = \matrix{-2 & 0 \\ 2 & -1} - \matrix{-1 & 0 \\0& -1 }=\matrix{ -1 & 0 \\ 2 & 0}\] This can be done as follows:
\[\begin{aligned}
\matrix{-1&0\\2&0\\}&\sim\matrix{1&0\\2&0\\}&{\blue{\begin{array}{c}-R_1\\\phantom{x}\end{array}}}\\\\ &\sim\matrix{1&0\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2-2R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = -1\) equals \(\left\{ r \cv{0\\1} \middle|\;r\in\mathbb R\right\}\).
If needed, we avoided here fractions in the solution.
In other words, the eigenspace of eigenvalue \(-1\) equals \(\left\langle\cv{0\\1}\right\rangle\)
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