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 \(\lambda = 5\) be an eigenvalue of the matrix \[A=\matrix{5 & 6 \\ 0 & 3}\] is. Then there must be a vector \(\vec{v}\) such that \(A\vec{v}=5\vec{v}\), that is, \[(A-5 I)\vec{v}=\vec{0}\text.\] In other words, we must find the kernel of the matrix \(A-5 I\).
We can do this through row reduction of the matrix \[A-5 I = \matrix{5 & 6 \\ 0 & 3} - \matrix{5 & 0 \\0& 5 }=\matrix{ 0 & 6 \\ 0 & -2}\] This can be done as follows:
\[\begin{aligned}
\matrix{0&6\\0&-2\\}&\sim\matrix{0&1\\0&-2\\}&{\blue{\begin{array}{c}{{1}\over{6}}R_1\\\phantom{x}\end{array}}}\\\\ &\sim\matrix{0&1\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2+2R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = 5\) equals \(\left\{ r \cv{1\\0} \middle|\;r\in\mathbb R\right\}\).
If needed, we avoided here fractions in the solution.
In other words, the eigenspace of eigenvalue \(5\) equals \(\left\langle\cv{1\\0}\right\rangle\)
We can do this through row reduction of the matrix \[A-5 I = \matrix{5 & 6 \\ 0 & 3} - \matrix{5 & 0 \\0& 5 }=\matrix{ 0 & 6 \\ 0 & -2}\] This can be done as follows:
\[\begin{aligned}
\matrix{0&6\\0&-2\\}&\sim\matrix{0&1\\0&-2\\}&{\blue{\begin{array}{c}{{1}\over{6}}R_1\\\phantom{x}\end{array}}}\\\\ &\sim\matrix{0&1\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2+2R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = 5\) equals \(\left\{ r \cv{1\\0} \middle|\;r\in\mathbb R\right\}\).
If needed, we avoided here fractions in the solution.
In other words, the eigenspace of eigenvalue \(5\) equals \(\left\langle\cv{1\\0}\right\rangle\)
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