Finding eigenvectors with eigenvalues

[Dusty912]

Homework Statement

So just curious about a specific problem that I am worries about running into on my test tomorrow. When trying to find eigen vectors with the eigen values what is there is a discrepancy between the two systems obtained after doing the matrix arithmetic?

such as after using the equation Av=λV and multiplying out all of the values, you end up with two equations that do not correspond. Such as getting y=0 for the top and 2x=3y for the second. is this even possible? and if it is what does it mean?

also looking for explanations for other scenarios such as y=-x and y=2x are the equations yielded. thank you very much for any help. and let me know if I need to go into more detail about what i am asking

 

[Buzz Bloom]

you end up with two equations that do not correspond

Hi Dusty:

Take a particular example, perhaps using a 2x2 matrix, and show the "two equations" that "do not correspond". Do you know how to find the two Eigen values for a 2x2 matrix?

Regards,
Buzz

 

[Dusty912]

well yes I do

 

[Dusty912]

by using det(A-λI)

 

[Buzz Bloom]

Hi Dusty:

OK. Make up a 2x2 matrix and find its two Eigen values. Then use that result as an example to show what you mean by
"you end up with two equations that do not correspond".

Regards,
Buzz

 

[Dusty912]

oh I see well let's say we have eigen values λ=1 and λ=-2 and we compute the eigen vectors with A= (4 2) for the top row and (1 1) for the bottom. Then using AV=λV, we obtain AV- λV=0 where vector V= (x,y) pluggin these values in and solving yield the top equation to be 4x+2y+1x=0 and the bottom being x+y +y=0. these were the two I was wondering about. because most of the time the same ration of y's and x's are achieved but i did run into one case where they did not, so is this even possible or will they always come out to the same ratio?

btw I did not actually compute the eigen values for this matrix, i just picked random numbers to get a discrepancy.

 

[Buzz Bloom]

Hi Dusty:

I confess do not understand the method you are using to find the Eigenvectors. It does not look right to me, but new ways to teach this stuff may have been developed since I learned this stuff many decades ago.

I suggest you take a look at

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors​

and specifically at the

3.1 Two dimensional example
​

Also, it you don't use the correct Eigenvalues, you won't get correct Eigenvectors, so I suggest writing down the quadratic equation in λ you get from

det(A-λI)=0.​

The two values of λ solving this equation are the Eigenvalues.

Regards,
Buzz

 

[HallsofIvy]

oh I see well let's say we have eigen values λ=1 and λ=-2 and we compute the eigen vectors with A= (4 2) for the top row and (1 1) for the bottom. Then using AV=λV, we obtain AV- λV=0 where vector V= (x,y) pluggin these values in and solving yield the top equation to be 4x+2y+1x=0 and the bottom being x+y +y=0. these were the two I was wondering about. because most of the time the same ration of y's and x's are achieved but i did run into one case where they did not, so is this even possible or will they always come out to the same ratio?

btw I did not actually compute the eigen values for this matrix, i just picked random numbers to get a discrepancy.

The whole point of "eigenvalues" is that the equation $AV= \lambda V$ has an infinite number of solutions. If you have the correct eigenvalues you can't get "two equations that conflict". If you do, then you have the wrong eigenvalues.