eigenvalues and eigenvectors

[PG1995]

Hi

Could you please help me with this query? If you can't see the image, then please use this username: imgshack4every1 and password: imgshack4every1. Please note that one of references does show how to find eigenvector but I wasn't able to understand the method. I think understanding it in the context of a simple linear system of equations would be the right way for me. Thank you.

Reference(s):
1: http://tutorial.math.lamar.edu/Classes/LinAlg/EVals_Evects.aspx
2: http://www.sosmath.com/matrix/eigen2/eigen2.html
3: https://www.electro-tech-online.com/custompdfs/2012/12/45answers.pdf

 

[dougy83]

see https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors -- the explanation is good, including a worked example.

Also https://mathworld.wolfram.com/Eigenvalue.html may include some info.

 

[PG1995]

Thank you, Doug.

I have been to Wikipedia page and have updated the system of equations in the linked document. I'm still struggling to understand the geometric meaning of eigenvalues and eigenvectors? Please have a look here. If you can't see the image, then please use this username: imgshack4every1 and password: imgshack4every1.

Regards
PG

PS: I'm still trying to understand it on my own. I can devote some more time to grasp it then probably it will make sense to me. Thank you for giving it a look but there is no need to reply to it now. I will update you if I need any help from your end. Many thanks.

 

[PG1995]

Hi

I believe this is what eigenvector and eigenvalue concept is all about. This is the link I refer to in the previous linked document.

Please also help me with this query. Thank you.

Regards
PG

 

[steveB]

Please also help me with this query. Thank you.

You can think of eigenvectors as directions (even "preferred" directions), even though they are not necessarily representative of directions in space. If you multiply an eigenvector by a scalar constant, it is still an eigenvector. Often, people prefer to scale the vector to have a magnitude of one. This is similar to defining a unit vector for directions in space.

 

[PG1995]

Thank you, Steve.

So, the calculator was scaling the eigenvectors to have a magnitude of one.

Best wishes
PG

 

[PG1995]

Hi

In the post #4 the eigenvector and corresponding eigenvalue for the matrix [1,0 ; 2,1] were found to be [0; 1] and "1" respectively. It means that when the transformation takes place the vector [0; 1] isn't affected and its direction and length are preserved. But wouldn't it be true for other vectors such as [0; 2], [0; 3] etc. which lie parallel to the eigenvector [0; 1]? If it's correct then what is so special about [0; 1] that it's the only one which is called 'eigenvector'? Please help me with it. Thank you.

Regards
PG

 

[steveB]

Hi

In the post #4 the eigenvector and corresponding eigenvalue for the matrix [1,0 ; 2,1] were found to be [0; 1] and "1" respectively. It means that when the transformation takes place the vector [0; 1] isn't affected and its direction and length are preserved. But wouldn't it be true for other vectors such as [0; 2], [0; 3] etc. which lie parallel to the eigenvector [0; 1]? If it's correct then what is so special about [0; 1] that it's the only one which is called 'eigenvector'? Please help me with it. Thank you.

Regards
PG

This is exactly why I mentioned that you can think of eigenvectors as analogous to "directions". You are exactly correct that any multiple of an eigenvector is also an eigenvector. People often like to normalize the eigenvector for this reason. The normalization method is arbitrary or a convention and often people prefer to scale the eigenvector to have a magnitude of 1.

 

[PG1995]

Thanks a lot.

Regards
PG