Eigenspaces homework problem

  • Thread starter g.lemaitre
  • Start date
  • #1
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Homework Statement


Find the Eigenspace of the following matrix:
[tex]\begin{bmatrix}
1 & 3 \\
4 & -3
\end {bmatrix}[/tex]
I'm skipping a few steps but the Eigenvalues are -5 and 3. Let's starts with -5. Skip a few more steps, I know I'm right, just trust me.
We now have the following matrix:
[tex]\begin{bmatrix}
-6 & -3 \\
-4 & -2
\end {bmatrix}[/tex]
Then you find the null space, which starts with putting it in reduced row echelon form:
[tex]\begin{bmatrix}
-6 & -3 \\
0 & 0
\end {bmatrix}[/tex]
you can reduce that further to
[tex]\begin{bmatrix}
-2 & -1 \\
0 & 0
\end {bmatrix}[/tex]
This is where I'm confused. This nullspace calculator http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi
says that the basis of the null space is
[tex]\begin{bmatrix}
-1 \\
2
\end {bmatrix}[/tex]
My textbook confirms that. How do I get from here
[tex]\begin{bmatrix}
-2 & -1 \\
0 & 0
\end {bmatrix}[/tex]
to there
[tex]\begin{bmatrix}
-1 \\
2
\end {bmatrix}[/tex]
I would think you would just eliminate the 2nd row and transpose the first row but that would give.
[tex]\begin{bmatrix}
-2 \\
-1
\end {bmatrix}[/tex]
 
Last edited:

Answers and Replies

  • #2
LCKurtz
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you can reduce that further to
[tex]\begin{bmatrix}
-2 & -1 \\
0 & 0
\end {bmatrix}[/tex]
This is where I'm confused. This nullspace calculator http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi
says that the basis of the null space is
[tex]\begin{bmatrix}
-1 \\
2
\end {bmatrix}[/tex]
My textbook confirms that. How do I get from here
[tex]\begin{bmatrix}
-2 & -1 \\
0 & 0
\end {bmatrix}[/tex]
to there
[tex]\begin{bmatrix}
-1 \\
2
\end {bmatrix}[/tex]
I would think you would just eliminate the 2nd row and transpose the first row but that would give.
[tex]\begin{bmatrix}
-2 \\
-1
\end {bmatrix}[/tex]

The matrix corresponds to -2x - y = 0 or y = -2x So$$
\begin{bmatrix}x\\ y \end{bmatrix}=\begin{bmatrix}x\\ -2x \end{bmatrix}
=x\begin{bmatrix}1\\ -2 \end{bmatrix} $$
Any nonzero constant works, so take ##x=-1## to get their answer.
 

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