Find the eigenvalues and eigenvectors for the matrix

In summary, the conversation discusses finding eigenvalues and eigenvectors for a given matrix, with a specific focus on the difference between two potential eigenvectors and how to discern between them. The conclusion is that, assuming the work is correct, both vectors are valid eigenvectors and it is unclear why a program like Mathematica would choose one over the other.
  • #1
tomeatworld
51
0

Homework Statement


Find the eigenvalues and eigenvectors for the matrix [{13,5},{2,4}]

Homework Equations


None

The Attempt at a Solution


Well eigenvalues is easy, and turn out to be 14 and 3.
So using eigenvalue 3, the two equations 10x1 + 5x2=0 and 2x1 + x2=0. Using these, I assumed the eigenvector to be [1,-2] but after putting it into Wolfram (or mathematica) it gives out the vector should be [-1,2]. Is there a difference? How should you discern between the two if not and where have I gone wrong if there is?
 
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  • #2


tomeatworld said:

Homework Statement


Find the eigenvalues and eigenvectors for the matrix [{13,5},{2,4}]

Homework Equations


None

The Attempt at a Solution


Well eigenvalues is easy, and turn out to be 14 and 3.
So using eigenvalue 3, the two equations 10x1 + 5x2=0 and 2x1 + x2=0. Using these, I assumed the eigenvector to be [1,-2] but after putting it into Wolfram (or mathematica) it gives out the vector should be [-1,2]. Is there a difference? How should you discern between the two if not and where have I gone wrong if there is?
Assuming your work is correct, there is no difference between <1, -2> and <-1, 2> as far as being eigenvectors. Each of these vectors is the -1 multiple of the other, so they are both in the same eigenspace, a subspace of dimension 1 (a line) in R2.
 
  • #3


Great. So do programs like mathematica choose them at random or is there a reason it chose <-1,2> over <1,-2>?
 
  • #5


Ok, thanks for the help!
 

1. What is the purpose of finding eigenvalues and eigenvectors for a matrix?

Finding eigenvalues and eigenvectors for a matrix is important in many areas of science, including physics, engineering, and computer science. It allows us to understand the behavior of a system or matrix, and can be used to solve systems of equations, perform transformations, and analyze data.

2. How do you find eigenvalues and eigenvectors for a matrix?

To find the eigenvalues and eigenvectors for a matrix, you first need to calculate the determinant of the matrix. Then, you solve the characteristic equation, which is a polynomial equation with the eigenvalues as the roots. Finally, you can use the eigenvalues to find the eigenvectors by solving a system of equations.

3. Can a matrix have more than one set of eigenvalues and eigenvectors?

Yes, a matrix can have multiple sets of eigenvalues and eigenvectors, as long as the eigenvalues are distinct. If two eigenvalues are the same, there may be infinitely many corresponding eigenvectors.

4. What is the relationship between eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are related in that the eigenvector is a vector that is scaled by the eigenvalue when multiplied by the original matrix. This means that the eigenvectors represent the directions in which the matrix has a simple scaling effect.

5. How are eigenvalues and eigenvectors used in data analysis?

Eigenvalues and eigenvectors are often used in data analysis to reduce the dimensionality of a dataset. This means that they can help us to identify the most important variables or features in a dataset, and to simplify the analysis and interpretation of the data.

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