Eigenvalues and eigenvectors of symmetric 2x2 matrix?

In summary, a general 2x2 symmetric matrix will not have an eigenvalue 0. The trace of a symmetric 2x2 matrix is equal to the sum of its eigenvalues. There is nothing special about the eigenvalues of a 2x2 symmetric matrix, but the eigenvalues of any symmetric matrix will always be real (non-imaginary). The eigenvalues of any 2x2 matrix will be its trace and 0 if and only if the determinant of the matrix vanishes. The statement that a 2x2 symmetric matrix can be diagonalized so that its eigenvalues are (trace) and 0 is not always true, as shown by the example of an identity matrix.
  • #1
malawi_glenn
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Hello

I recall, I think, that there is a lemma which states that a 2x2 symmetric matrix can be diagonalized so that its eigenvalues are (trace) and 0.

I can not find it anywhere =/ I think it was a physics teacher who told us this a couple of years ago, can anyone enlighten me?

cheers
 
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  • #2
A general 2x2 symmetric matrix will not have an eigenvalue 0. The trace of a symmetric 2x2 matrix is equal to the sum of it's eigenvalues, maybe that was what you were thinkig of?
 
Last edited:
  • #3
dx said:
A general 2x2 symmetric matrix will not have an eigenvalue 0. The trace of a symmetric 2x2 matrix is equal to the sum of it's eigenvalues, maybe that was what you were thinkig of?

hmm yeah, maybe something like that.

Thanx for input
 
  • #4
The trace of a matrix is always equal to the sum of it's eigenvalues. I don't think there is anything special about the eigenvalues of a 2x2 symmetric matrix, but eigen values of any symmetric matrix will be real (non-imaginary).
 
  • #5
The eigenvalues of any 2x2 matrix will be its trace and 0 if and only if the determinant of the matrix vanishes
 
  • #6
<<I recall, I think, that there is a lemma which states that a 2x2 symmetric matrix can be diagonalized so that its eigenvalues are (trace) and 0>>

We can see that is not true by considering a trivial example of a matrix that already is diagonalized.

For example, if you digonalize an identity matrix, you get back the identity matrix. It, of course, does not have (trace) and zero as the diagonal elements.
 

1. What are eigenvalues and eigenvectors of a symmetric 2x2 matrix?

Eigenvalues and eigenvectors are important concepts in linear algebra that are used to understand the properties of a square matrix. In the case of a symmetric 2x2 matrix, the eigenvalues are the two values that, when multiplied by the corresponding eigenvectors, produce the original matrix.

2. How do you calculate the eigenvalues and eigenvectors of a symmetric 2x2 matrix?

To calculate the eigenvalues of a symmetric 2x2 matrix, you can use the quadratic formula. Once you have the eigenvalues, you can find the eigenvectors by solving the corresponding system of equations using the eigenvalues.

3. What do eigenvalues and eigenvectors represent in a symmetric 2x2 matrix?

The eigenvalues of a symmetric 2x2 matrix represent the stretching or shrinking factor along the corresponding eigenvectors. In other words, they indicate the direction and magnitude of the transformation that the matrix performs on the eigenvectors.

4. Can a symmetric 2x2 matrix have complex eigenvalues and eigenvectors?

Yes, a symmetric 2x2 matrix can have complex eigenvalues and eigenvectors. This is because the eigenvalues and eigenvectors of a matrix are found by solving a system of equations, and in some cases, the solutions may be complex numbers.

5. How can eigenvalues and eigenvectors of a symmetric 2x2 matrix be used in real-world applications?

Eigenvalues and eigenvectors of a symmetric 2x2 matrix have various applications in fields such as physics, engineering, and computer science. They are used to analyze and model systems that involve linear transformations, such as in image processing, signal processing, and quantum mechanics.

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