Eigenvalues and Normalised Eigenvectors

In summary, to find the eigenvalues and normalized eigenvectors of a matrix, you can solve for the eigenvalues by setting the determinant of (H - λI) equal to zero and then plugging them into the equation (H - λI)x = 0 to get the eigenvectors. Normalizing these eigenvectors involves dividing them by their magnitude, or unit length. In the given example, the eigenvalues were (h±g) and the corresponding normalized eigenvectors were (1,1) and (1,-1).
  • #1
captainjack2000
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0

Homework Statement


I have a matrix
H= [h g
g h]
and I need to find the eigenvalues and normalised eigenvectors


Homework Equations





The Attempt at a Solution


I subtracted lamda from the diagonal and then solved for the determinant equally zero. The eigenvalues I found were
(h-lambda)^2=g^2
so (h-lambda)=+/- g
lamdba=h+/-g

but I'm not sure how to find the normalised eigenvectors?
 
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  • #2
After finding the eigenvalues, plug them back into the equation (A - λ I)x = 0, one by one, to get your eigenvectors. Then normalize them.
 
  • #3
Thank you very much.

So what I have done is
for eigenvalue h+g I have two equations
hx+gy=hx+gx and gx+hy = hy+gy which gives x=y so eigenvalue h+g has eigenvector (1,1)
and for eigenvalue h-g I have two equations
hx+gy=hx-gx and hx+gy=hy-gy so x=-y so eigenvalue h-g has eigenvector (1,-1)

Is that correct?
 
  • #4
are these normalised?
 
  • #5
What does it mean for a vector to be normalized?
 
  • #6
to be of unit length...which these are right?
 
  • #7
I am not going to check your calculations, but with respect to the standard norm neither (1, 1) nor (1, -1) have unit length, since |(1, 1)|=(|(1, -1)|) = √2.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts that are used to analyze linear transformations, such as those found in physics, engineering, and computer graphics. Eigenvalues represent the scaling factor of the eigenvectors, which are the special set of vectors that do not change their direction when a transformation is applied to them.

2. How are eigenvalues and eigenvectors related?

Eigenvalues and eigenvectors are related through a linear transformation. When a linear transformation is applied to an eigenvector, the resulting vector is a scaled version of the original eigenvector, and the scaling factor is the corresponding eigenvalue. In other words, the eigenvector and eigenvalue are paired together.

3. What is the significance of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important because they allow us to simplify and analyze complex linear transformations. They can help us understand the behavior of systems and predict future outcomes. They are also used in various applications, such as image compression and data analysis.

4. How are eigenvalues and eigenvectors calculated?

To calculate the eigenvalues and eigenvectors of a matrix, we first need to find the characteristic polynomial of the matrix. This polynomial is then solved to find the eigenvalues, which are the roots of the polynomial. The corresponding eigenvectors can be found by substituting the eigenvalues into the original matrix equation and solving for the eigenvectors.

5. What are normalised eigenvectors?

Normalised eigenvectors are eigenvectors that have been scaled to have a length of 1. This is done by dividing each element of the eigenvector by its magnitude. Normalised eigenvectors are useful because they simplify calculations and can help in visualizing the direction and magnitude of the transformation represented by the eigenvector.

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