Eigenvalues and eigenvectors of symmetric 2x2 matrix?

[malawi_glenn]

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

 

[dx]

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?

 

[malawi_glenn]

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

 

[daviddoria]

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).

 

[kjarunkj]

The eigenvalues of any 2x2 matrix will be its trace and 0 if and only if the determinant of the matrix vanishes

 

[ApplePion]

<<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.