How do I find the eigenvalues and eigenvectors of a given matrix?

In summary, The homework statement is trying to find the eigenvalues and eigenvectors of a matrix. The Attempt at a Solution found the eigenvalues and eigenvectors of a matrix, but was having trouble finding the eigenvectors for all the other values.
  • #1
LoopQG
22
0

Homework Statement



Find the eigenvalues and eigenvectors of this matrix.

[4 0 1/2 ]
[0 -5 0 ]
[1/2 0 1 ]

3. The Attempt at a Solution

I have found the eigenvalues = -5, 5/2 + sqrt(5/2), 5/2 - sqrt(5/2)

I know to get the eigenvectors you subtract the eigenvalues from thediagonal and set each row equal to zero and solve for the 3 components.

for -5

[9 0 1/2]
[0 0 0]
[1/2 0 -5]

so 9x +1/2 z = 0
1/2x -5z=0

and y =0

but i get zero for all components even when i do the other eigenvalues. can anyone show me what i am doing wrong?
 
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  • #2
Hi LoopQG! :smile:
LoopQG said:
[9 0 1/2]
[0 0 0]
[1/2 0 -5]

so 9x +1/2 z = 0
1/2x -5z=0

and y =0

nooo :redface:

your equations don't mention y, so y can be anything!

(and isn't it obvious that the eigenvectors must be [0,y,0]? :wink:)
 
  • #3
Yes but what about the vectors for the other values. I still get all the components equal zero. I have plugged it into MATLAB and i know what the other eigenvectors are but i don't see how to get them analytically
 
  • #4
For the first and third rows, for eigenvalue -5, you got two different equations for x and z, so they could only be satisfied by x = z = 0.

But for either of the other eigenvalues, the equations for the first and third rows should be the same (hint: (3 + √10)(3 - √10) = -1 :wink:).
 
  • #5
ok so after figuring out i had to normalize i first chose x=1 so z=-3 +sqrt(10) for 5/2 + sqrt(5/2)

and for 5/2 - sqrt(5/2) i set z=1 and x=3-sqrt(10)

and y=0 for both

can you verify if this is correct? thank you for all the guidance.
 
  • #6
Hi LoopQG! :smile:

Yes, (1, 0, -3+√10) and (3-√10, 0, 1) look fine.

But there are other ways which you might think are neater …

eg √10 > 3, so you could write them (1, 0, √10 - 3) and (√10 - 3, 0, -1)

or you could even use (√10 - 3)(√10 + 3) = 1, and start both of them with x = ±1. :wink:
 
  • #7
excellent. hanks so much for your help. I do have one more question though.

The idea behind this was to find the principal axes of the strain rate tensor, which is the matrix in post 1. And find the principal rates of strain. For the principal rates of strain i am getting from my text that I should get a diagonal 3x3 with the eigenvalues on the diagonal.

the strain tensor should trans form like E(principal rate of strain) = C E(original strain tensor) C (transpose)

where C is the matrix of all 3 eigenvectors.

When I use the eigenvectors i have calculated i do not get a diagonal matrix, is there a certain method i should use for normalizing the eigenvectors to get the principal axes or am I just doing the algebra wrong.

Thanks again for all the clarification this has been a good refresher in linear algebra.
 
  • #8
Hi LoopQG! :smile:
LoopQG said:
… For the principal rates of strain i am getting from my text that I should get a diagonal 3x3 with the eigenvalues on the diagonal.

the strain tensor should trans form like E(principal rate of strain) = C E(original strain tensor) C (transpose)

where C is the matrix of all 3 eigenvectors.

ah, no, it's not CT, it's C-1

see http://en.wikipedia.org/wiki/Jordan_normal_form#Example :wink:
 

Related to How do I find the eigenvalues and eigenvectors of a given matrix?

1. What are eigenvectors and why are they important in science?

Eigenvectors are special vectors that represent the directions along which a linear transformation acts by simply scaling the vector. They are important in science because they can help us understand and analyze complex systems, such as in physics, engineering, and data analysis.

2. How do you find eigenvectors?

To find the eigenvectors of a matrix, you first need to find the eigenvalues by solving the characteristic equation. Once you have the eigenvalues, you can then plug them back into the original matrix to find the corresponding eigenvectors. This process can be done by hand or with the help of software, such as MATLAB or Python.

3. What is the difference between eigenvectors and eigenvalues?

Eigenvectors are vectors that represent the direction of a linear transformation, while eigenvalues are the corresponding scalar values that represent how much the eigenvectors are scaled by the transformation. In other words, eigenvectors show the direction of change while eigenvalues show the magnitude of change.

4. Can you have multiple eigenvectors for the same eigenvalue?

Yes, it is possible to have multiple eigenvectors for the same eigenvalue. This is because eigenvectors are not unique and can be scaled by any nonzero constant and still remain eigenvectors. However, the number of linearly independent eigenvectors for a given eigenvalue is limited by the dimension of the matrix.

5. How are eigenvectors used in real-world applications?

Eigenvectors are used in various real-world applications, such as image and signal processing, quantum mechanics, and machine learning. They can help identify patterns and relationships in data, reduce the dimensionality of data, and solve complex equations and systems. They are also used in the development of algorithms for optimization and data compression.

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