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 \[A=\matrix{1 & -6 \\ 1 & -4}\] 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 \[A+I = \matrix{1 & -6 \\ 1 & -4} - \matrix{-1 & 0 \\0& -1 }=\matrix{ 2 & -6 \\ 1 & -3}\] This can be done as follows:
\[\begin{aligned}
\matrix{2&-6\\1&-3\\}&\sim\matrix{1&-3\\2&-6\\}&{\blue{\begin{array}{c}R_2\\R_1\end{array}}}\\\\ &\sim\matrix{1&-3\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2-2R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = -1\) equals \(\left\{ r \cv{3\\1} \middle|\;r\in\mathbb R\right\}\).
If needed, we avoided here fractions in the solution.
In other words, the eigenspace of eigenvalue \(-1\) equals \(\left\langle\cv{3\\1}\right\rangle\)
We can do this through row reduction of the matrix \[A+I = \matrix{1 & -6 \\ 1 & -4} - \matrix{-1 & 0 \\0& -1 }=\matrix{ 2 & -6 \\ 1 & -3}\] This can be done as follows:
\[\begin{aligned}
\matrix{2&-6\\1&-3\\}&\sim\matrix{1&-3\\2&-6\\}&{\blue{\begin{array}{c}R_2\\R_1\end{array}}}\\\\ &\sim\matrix{1&-3\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2-2R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = -1\) equals \(\left\{ r \cv{3\\1} \middle|\;r\in\mathbb R\right\}\).
If needed, we avoided here fractions in the solution.
In other words, the eigenspace of eigenvalue \(-1\) equals \(\left\langle\cv{3\\1}\right\rangle\)
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