How do you find this eigenvector?

In summary, an eigenvector is a vector that remains unchanged when multiplied by a specific matrix and is associated with a scalar called the eigenvalue. Eigenvectors are important in various fields of mathematics and science, including linear algebra and quantum mechanics, as they can simplify complex systems and solve eigenvalue problems. To find an eigenvector, one must first find the eigenvalues of the matrix and then use them to solve a system of equations or apply computational methods. The dominant eigenvector, corresponding to the largest eigenvalue, represents the direction of greatest change in a system and can be used to identify important features or patterns in data analysis and machine learning. While there can be infinitely many eigenvectors for a single eigenvalue, there
  • #1
msell2
15
0
[0 1]
[-2 -2] This is the 2x2 matrix.

[λ -1]
[2 λ+2] This is the matrix that equals λI - A.

Here are the eigenvalues I found:
λ = -1 + i, -1 - i

I am really confused at what to do next to find the eigenvectors. I keep looking online for help but I still can't figure it out...

Thanks!
 
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  • #2
msell2 said:
[0 1]
[-2 -2] This is the 2x2 matrix.

[λ -1]
[2 λ+2] This is the matrix that equals λI - A.

Here are the eigenvalues I found:
λ = -1 + i, -1 - i

I am really confused at what to do next to find the eigenvectors. I keep looking online for help but I still can't figure it out...

Thanks!

Now put one of your eigenvalues into λI - A and try to find a vector v such that (λI - A)v=0. Then v would be an eigenvector of λ. It's just a set of linear equations to solve with a free parameter.
 
  • #3
msell2 said:
[0 1]
[-2 -2] This is the 2x2 matrix.

[λ -1]
[2 λ+2] This is the matrix that equals λI - A.

Here are the eigenvalues I found:
λ = -1 + i, -1 - i

I am really confused at what to do next to find the eigenvectors. I keep looking online for help but I still can't figure it out...

Thanks!

For ##\lambda## equal an eigenvalue, your two equations are dependent, so you only need to solve one and you have one free variable. The first one is the same as ##\lambda x - y =0## or ##y = \lambda x##. So let ##x = 1## and what do you get for ##y##? Do that for each value of ##\lambda## and you will have two [x,y] eigenvectors.
 
  • #4
Dick said:
Now put one of your eigenvalues into λI - A and try to find a vector v such that (λI - A)v=0. Then v would be an eigenvector of λ. It's just a set of linear equations to solve with a free parameter.
I get what I need to do in theory, I just don't actually know how to do it. How do you find v such that (λI-A)v=0?
 
  • #5
msell2 said:
I get what I need to do in theory, I just don't actually know how to do it. How do you find v such that (λI-A)v=0?

Just try it! What is the matrix (λI-A) when λ=-1+i? Write that matrix times a vector (x,y), set it equal to zero and try to find a solution for x and y.
 

FAQ: How do you find this eigenvector?

What is an eigenvector?

An eigenvector is a vector that does not change direction when multiplied by a particular matrix. It corresponds to a scalar value known as the eigenvalue, which represents the amount by which the eigenvector is stretched or compressed by the matrix.

Why are eigenvectors important?

Eigenvectors are important in many areas of mathematics and science, particularly in linear algebra and quantum mechanics. They can be used to simplify complex systems and solve eigenvalue problems, and they have a wide range of applications in fields such as computer graphics, data analysis, and physics.

How do you find an eigenvector?

To find an eigenvector, you first need to find the eigenvalues of the matrix. This can be done by solving the characteristic equation for the matrix. Once you have the eigenvalues, you can use them to find the corresponding eigenvectors by solving a system of equations involving the matrix and the eigenvalue. Alternatively, you can use computational methods such as the power method or the QR algorithm to find the eigenvectors.

What is the significance of the eigenvector corresponding to the largest eigenvalue?

The eigenvector corresponding to the largest eigenvalue is known as the dominant eigenvector and it represents the direction of greatest change in a system. In applications such as data analysis and machine learning, the dominant eigenvector can be used to identify the most important features or patterns in a dataset.

Can there be multiple eigenvectors for a single eigenvalue?

Yes, there can be multiple eigenvectors for a single eigenvalue. In fact, there are infinitely many eigenvectors for any given eigenvalue. However, in some cases, there may only be one linearly independent eigenvector corresponding to a particular eigenvalue.

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