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 \(\lambda = 3\) be an eigenvalue of the matrix \[A=\matrix{3 & 10 \\ 0 & -2}\] is. Then there must be a vector \(\vec{v}\) such that \(A\vec{v}=3\vec{v}\), that is, \[(A-3 I)\vec{v}=\vec{0}\text.\] In other words, we must find the kernel of the matrix \(A-3 I\).
We can do this through row reduction of the matrix \[A-3 I = \matrix{3 & 10 \\ 0 & -2} - \matrix{3 & 0 \\0& 3 }=\matrix{ 0 & 10 \\ 0 & -5}\] This can be done as follows:
\[\begin{aligned}
\matrix{0&10\\0&-5\\}&\sim\matrix{0&1\\0&-5\\}&{\blue{\begin{array}{c}{{1}\over{10}}R_1\\\phantom{x}\end{array}}}\\\\ &\sim\matrix{0&1\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2+5R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = 3\) 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 \(3\) equals \(\left\langle\cv{1\\0}\right\rangle\)
We can do this through row reduction of the matrix \[A-3 I = \matrix{3 & 10 \\ 0 & -2} - \matrix{3 & 0 \\0& 3 }=\matrix{ 0 & 10 \\ 0 & -5}\] This can be done as follows:
\[\begin{aligned}
\matrix{0&10\\0&-5\\}&\sim\matrix{0&1\\0&-5\\}&{\blue{\begin{array}{c}{{1}\over{10}}R_1\\\phantom{x}\end{array}}}\\\\ &\sim\matrix{0&1\\0&0\\}&{\blue{\begin{array}{c}\phantom{x}\\R_2+5R_1\end{array}}} \end{aligned}\] So the eigenspace for \(\lambda = 3\) 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 \(3\) equals \(\left\langle\cv{1\\0}\right\rangle\)
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