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Real and Zero eigenvalues

  • #1

Homework Statement


[tex]
\frac{d\vec{Y}}{dt}
=
\begin{bmatrix}
2 & 4 \\
3 & 6
\end{bmatrix}
\vec{Y}[/tex]
Find the eigenvalues and eigenvectors

Homework Equations




The Attempt at a Solution


I found the eigenvectors to be
[tex]
\vec{v_1} =
\begin{bmatrix}
2 \\
1
\end{bmatrix}
,
\vec{v_2} =
\begin{bmatrix}
2 \\
-3
\end{bmatrix}
[/tex]

I found a widget on Wolfram Alpha that says the second eigenvector should be:
[tex]
\begin{bmatrix}
2 \\
3
\end{bmatrix}
[/tex]

I am more inclined to believe wolfram alpha is correct, but can someone show me why?
 

Answers and Replies

  • #2
pasmith
Homework Helper
1,738
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How are we to determine whether, and if so how, you have made an error if you don't post your working?
 
  • #3
It was mostly by looking at the matrix and trying to make them the same and now that you say that when I do the algebra on it to post it, I see what I did wrong. I just tried to look at 3 and -2 to find the solution when I should have said 3x-2y=0 and 3x=2y then the vector becomes clear.
 
  • #4
Ray Vickson
Science Advisor
Homework Helper
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It was mostly by looking at the matrix and trying to make them the same and now that you say that when I do the algebra on it to post it, I see what I did wrong. I just tried to look at 3 and -2 to find the solution when I should have said 3x-2y=0 and 3x=2y then the vector becomes clear.
Neither of your ##v_1## or ##v_2## are eigenvectors. If ##A## is your matrix we have ##Av_1 = (8,12)^T##, which cannot be a multiple of ##v_1##: the first component of ##v_1## is larger than the second component, while the opposite is true for ##Av_1##. Similarly, ##v_2## cannot be an eigenvector of ##A## because the components of ##v_2## have opposite signs, while those of ##Av_2## have the same sign.

Also: you did not show the eigenvalues.
 
  • #5
Eigens are 0,8. The problem was v2 I ommited a negative sign in the first vector on accident.
 
  • #6
Ray Vickson
Science Advisor
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Eigens are 0,8. The problem was v2 I ommited a negative sign in the first vector on accident.
I do not understand what you are trying to say. What is the correct ##v_1##? What is the correct ##v_2##? Instead of trying to describe these in words, just show the actual numerical entries.
 
  • #7
[tex]
\vec{v_1}=
\begin{bmatrix}
-2 \\
1
\end{bmatrix}
,
\vec{v_2}=
\begin{bmatrix}
2 \\
3
\end{bmatrix}
[/tex]

Sory for putting so little work into this, I've got the answer and moved onto the next problem at this point.
 

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