Question

Consider matrix A
(1 2 1)
(6 -1 0)
(-1 -2 -1)
Find the eigenvalues and the eigenvectors.

I already found the eigenvalues to be -4, 0, 3.
I am having trouble however finding the eigenvectors. Can someone show me how to solve for the eigenvectors? Any help would be appreciated.

Answer 1

To get an eigenvector, you solve for \(\overrightarrow{x}\) in \((\lambda I-A)\overrightarrow{x}=\overrightarrow{0}\). \(I\) is the identity matrix and \(A\) is your matrix

Answer 2

So to find the corresponding eigenvector for -4, we have:
\[
\left(
\begin{bmatrix}
-4 & 0 & 0 \\
0 & -4 & 0 \\
0 & 0 & -4
\end{bmatrix}
-
\begin{bmatrix}
1 & 2 & 1 \\
6 & -1 & 0 \\
-1 & -2 & -1
\end{bmatrix}
\right)
\overrightarrow{x}
=
\overrightarrow{0}
\]

Answer 3

Just do matrix subtraction, then do elimination or whatever to solve for \(\overrightarrow{x}\).

Answer 4

http://tutorial.math.lamar.edu/Classes/LinAlg/EVals_Evects.aspx

Answer 5

Note that eigenvectors are magnitude independent