051 how they got the eigenspaces ?

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In summary, the conversation discussed the eigenspaces of a matrix with values of -1 and 2. The eigenvalues were determined to be 3 and 4. The corresponding eigenvectors were found to be (2, 3) and any multiple of it.
  • #1
karush
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ok I didn't understand how they got the eigenspaces

the original matrix was
$A=\left[\begin{array}{rrr}−1&2\\−6&6\end{array} \right]$
so think I got values correct $\lambda=2,3$

https://dl.orangedox.com/wlKD7eKSWiQ79alYD6
 
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  • #2
Yes, the eigenvalues are 3 and 4.

Any eigenvector corresponding to eigenvalue 3 must satisfy
$\begin{bmatrix}-1 & 2 \\ -6 & 6\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}3x \\ 3y \end{bmatrix}$
so -x+ 2y= 3x and -6x+6y= 3y. What must x and y be? (Those two equations reduce to the same thing so there are an infinite number of solutions- a one dimensional subspace.)

Any eigenvector corresponding to eigenvalue 4 must satisfy
$\begin{bmatrix}-1 & 2 \\ -6 & 6\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}4x \\ 4y \end{bmatrix}$
so -x+ 2y= 4x and -6x+6y= 4y. What must x and y be?

Those eigenvectors span the "eigenspace".
 
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  • #3
Country Boy said:
Any eigenvector corresponding to eigenvalue 4 must satisfy
$\begin{bmatrix}-1 & 2 \\ -6 & 6\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}4x \\ 4y \end{bmatrix}$
so -x+ 2y= 4x and -6x+6y= 4y. What must x and y be?

Those eigenvectors span the "eigenspace".
That last is wrong because, of course, the eigenvalue was 2, not 4!
Instead -x+ 2y= 2x and -6x+ 6y= 2y.
2y= 3x and -6x= -4y both reduce to 3x= 2y. In particular, if we take x= 2, then 3x= 6= 2y, y= 3. One eigenvector is (2, 3) but any multiple is also an eigenvector.
 

Related to 051 how they got the eigenspaces ?

1. What are eigenspaces?

Eigenspaces are subspaces of a vector space that are associated with a specific eigenvalue of a linear transformation. They are composed of all the eigenvectors corresponding to that eigenvalue.

2. How do you find eigenspaces?

To find eigenspaces, you first need to find the eigenvalues of a linear transformation. Then, for each eigenvalue, you can solve for the corresponding eigenvectors. These eigenvectors make up the eigenspace for that specific eigenvalue.

3. What are the applications of eigenspaces?

Eigenspaces have various applications in mathematics and science, including in linear algebra, differential equations, and quantum mechanics. They are also useful in data analysis and machine learning.

4. Can eigenspaces have a dimension greater than 1?

Yes, eigenspaces can have a dimension greater than 1. In fact, the dimension of an eigenspace is equal to the multiplicity of the corresponding eigenvalue, which can be greater than 1.

5. How are eigenspaces related to eigenvalues?

Eigenspaces and eigenvalues are closely related. Eigenspaces are composed of all the eigenvectors associated with a specific eigenvalue. Additionally, the dimension of an eigenspace is equal to the multiplicity of the corresponding eigenvalue.

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