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# What is an Eigenframe

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1. Can anyone tell me what an eigenframe is? Google doesn't return much. I know of eigenvectors, eigenvalues and eigenspaces but not eigenframes.
2. If anyone knows, this sounds like something pretty cool.
3. (Original post by djpailo)
Can anyone tell me what an eigenframe is? Google doesn't return much. I know of eigenvectors, eigenvalues and eigenspaces but not eigenframes.
Would probably be helpful if you gave us the context in which you've seen it being used in.
4. (Original post by djpailo)
I believe that you are referring to an Einsteinian concept related to tensors and covariance matrices. Maybe searching for those will lead you closer.
5. (Original post by djpailo)
Can anyone tell me what an eigenframe is? Google doesn't return much. I know of eigenvectors, eigenvalues and eigenspaces but not eigenframes.
This is not terminology that I've come across before, but it appears to be used as follows. Any symmetric matrix has a basis in which it is diagonal - and this basis can be formed from the eigenvectors of the matrix - such a basis is called an eigenframe for the matrix and hence for the tensor corresponding to it. Another name used in the literature is a "principal axes frame".
6. (Original post by Gregorius)
This is not terminology that I've come across before, but it appears to be used as follows. Any symmetric matrix has a basis in which it is diagonal - and this basis can be formed from the eigenvectors of the matrix - such a basis is called an eigenframe for the matrix and hence for the tensor corresponding to it. Another name used in the literature is a "principal axes frame".
So does this mean your representing the matrix using a different basis, and under this basis, you've made the matrix diagonal and now we have different eigvenvalues for the matrix?

Thanks for mentioning principal axes frame. I wonder if this refers to principle axes of strain (or stress?) etc that I faintly recollect from structures?
7. (Original post by djpailo)
So does this mean your representing the matrix using a different basis, and under this basis, you've made the matrix diagonal and now we have different eigvenvalues for the matrix?
Yes, the matrix is represented in a new basis and it is diagonal in this basis. However, the eigenvalues remain the same - these are invariants of the tensor represented by the matrix. Moreover, if the tensor/matrix varies smoothly from point to point, you'll be able to choose a smooth change of basis that diagonalizes the tensor/metrix at every point. (There'll be some conditions, but I've forgotten them!).

Thanks for mentioning principal axes frame. I wonder if this refers to principle axes of strain (or stress?) etc that I faintly recollect from structures?
That's going well beyond my area, but it sounds plausible.
8. (Original post by Gregorius)
Yes, the matrix is represented in a new basis and it is diagonal in this basis. However, the eigenvalues remain the same - these are invariants of the tensor represented by the matrix. Moreover, if the tensor/matrix varies smoothly from point to point, you'll be able to choose a smooth change of basis that diagonalizes the tensor/metrix at every point. (There'll be some conditions, but I've forgotten them!).

That's going well beyond my area, but it sounds plausible.
Okay thanks.

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