What is the Eigenspace for a 2x2 matrix with eigenvalues -5 and 3?

In summary, the conversation discusses finding the Eigenspace of a given matrix, which involves finding the null space. The steps to find the null space include putting the matrix in reduced row echelon form and then using a nullspace calculator to find the basis of the null space. The final result is a vector that can be multiplied by any nonzero constant to get the appropriate answer.
  • #1
g.lemaitre
267
2

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]
 
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  • #2


g.lemaitre said:
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.
 

What is an eigenspace?

An eigenspace is a subspace of a vector space that consists of all the eigenvectors associated with a particular eigenvalue. It is a fundamental concept in linear algebra and is used to solve systems of linear equations and understand the behavior of linear transformations.

How do I determine the dimension of an eigenspace?

The dimension of an eigenspace is equal to the multiplicity of the associated eigenvalue. This can be found by calculating the algebraic multiplicity of the eigenvalue (the number of times it appears as a root of the characteristic polynomial) and comparing it to the geometric multiplicity (the dimension of the corresponding eigenspace).

How do I find the basis for an eigenspace?

To find the basis for an eigenspace, you first need to find the eigenvalues for the given matrix or linear transformation. Then, for each eigenvalue, you can find the associated eigenvectors by solving the system of equations (A - λI)x = 0, where A is the matrix and λ is the eigenvalue. The set of all eigenvectors for a particular eigenvalue forms a basis for the eigenspace associated with that eigenvalue.

Can an eigenspace be empty?

Yes, it is possible for an eigenspace to be empty. This occurs when the eigenvalue associated with the eigenspace has a multiplicity of 0, meaning there are no eigenvectors associated with it. In this case, the eigenspace does not exist.

What are some applications of eigenspaces?

Eigenspaces have many real-world applications, including in computer graphics, data compression, and quantum mechanics. They are also used in solving differential equations and understanding the behavior of dynamical systems. Additionally, eigenspaces are used in various machine learning algorithms for data analysis and pattern recognition.

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