Finding the eigenvectors for T()

1. The problem statement, all variables and given/known data

Which of the following is not an eigenvector for [tex]
T \left(
\left[ {\begin{array}{cc}
x \\
y \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
x + y \\
x+ y \\
\end{array} } \right]
[/tex] ?

A) v = [-2 -2]T
B) v = [1 -1]T
C) v = [1 2]T
D) All are eigenvectors

2. Relevant equations

Ax = [tex]\lambda[/tex]x

3. The attempt at a solution
My problem is that all eigenvectors I've computed have come from 2x2 matrices. My best guess on starting is
[tex]
T \left(
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
-4 \\
-4 \\
\end{array} } \right]
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right]
=
[/tex]

but this obviously doesn't work because of the size. How do I find an eigenvalue of a 2x1 matrix? Is it possible? Am I even looking at this correctly?

Edit: I think I've figured it out. First, I constructed the standard matrix for T and got [tex]\left[ {\begin{array}{cc}
1&1 \\
1&1 \\
\end{array} } \right] [/tex]
Then I used Ax = [tex]\lambda[/tex]x with the vectors given to find the eigenvalues. Letter C didn't have an eigenvalue, so that is the answer.
[tex]
T \left(
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
1&1 \\
1&1 \\
\end{array} } \right]
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right] =
\left[ {\begin{array}{cc}
-4 \\
-4 \\
\end{array} } \right] = 2
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right]
[/tex]
[tex]
T \left(
\left[ {\begin{array}{cc}
1 \\
-1 \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
1&1 \\
1&1 \\
\end{array} } \right]
\left[ {\begin{array}{cc}
1 \\
-1 \\
\end{array} } \right] =
\left[ {\begin{array}{cc}
0 \\
0 \\
\end{array} } \right] = 0
\left[ {\begin{array}{cc}
1 \\
-1 \\
\end{array} } \right]
[/tex]
[tex]
T \left(
\left[ {\begin{array}{cc}
1 \\
2 \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
1&1 \\
1&1 \\
\end{array} } \right]
\left[ {\begin{array}{cc}
1 \\
2 \\
\end{array} } \right] =
\left[ {\begin{array}{cc}
3 \\
3 \\
\end{array} } \right] = ???
[/tex]
 
Last edited:
31,932
3,895
1. The problem statement, all variables and given/known data

Which of the following is not an eigenvector for [tex]
T \left(
\left[ {\begin{array}{cc}
x \\
y \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
x + y \\
x+ y \\
\end{array} } \right]
[/tex] ?

A) v = [-2 -2]T
B) v = [1 -1]T
C) v = [1 2]T
D) All are eigenvectors

2. Relevant equations

Ax = [tex]\lambda[/tex]x

3. The attempt at a solution
My problem is that all eigenvectors I've computed have come from 2x2 matrices. My best guess on starting is
[tex]
T \left(
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
-4 \\
-4 \\
\end{array} } \right]
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right]
=
[/tex]

but this obviously doesn't work because of the size. How do I find an eigenvalue of a 2x1 matrix? Is it possible? Am I even looking at this correctly?

Edit: I think I've figured it out. First, I constructed the standard matrix for T and got [tex]\left[ {\begin{array}{cc}
1&1 \\
1&1 \\
\end{array} } \right] [/tex]
Then I used Ax = [tex]\lambda[/tex]x with the vectors given to find the eigenvalues. Letter C didn't have an eigenvalue, so that is the answer.
[tex]
T \left(
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
1&1 \\
1&1 \\
\end{array} } \right]
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right] =
\left[ {\begin{array}{cc}
-4 \\
-4 \\
\end{array} } \right] = 2
\left[ {\begin{array}{cc}
-2 \\
-2 \\
\end{array} } \right]
[/tex]
[tex]
T \left(
\left[ {\begin{array}{cc}
1 \\
-1 \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
1&1 \\
1&1 \\
\end{array} } \right]
\left[ {\begin{array}{cc}
1 \\
-1 \\
\end{array} } \right] =
\left[ {\begin{array}{cc}
0 \\
0 \\
\end{array} } \right] = 0
\left[ {\begin{array}{cc}
1 \\
-1 \\
\end{array} } \right]
[/tex]
[tex]
T \left(
\left[ {\begin{array}{cc}
1 \\
2 \\
\end{array} } \right] \right) =
\left[ {\begin{array}{cc}
1&1 \\
1&1 \\
\end{array} } \right]
\left[ {\begin{array}{cc}
1 \\
2 \\
\end{array} } \right] =
\left[ {\begin{array}{cc}
3 \\
3 \\
\end{array} } \right] = ???
[/tex]
Your answer is correct, but you have made extra work for yourself in this problem. You didn't need to find a matrix representation for this transformation. All you needed to do was determine whether T(x) is equal to a scalar multiple of x.
For example, T(-2, -2) = (-4, -4) = 2*(-2, -2).
 

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