The discussion centers on how to represent tensors as matrices, particularly in the context of general relativity (GR). Participants explore the conversion of different ranks of tensors into matrix forms and the implications of introducing bases for these representations.
Participants express varying levels of understanding regarding the representation of tensors as matrices, with some agreement on the utility of rank-2 tensors in GR, but no consensus on the clarity of the conversion process.
There are indications that the discussion may lack sufficient detail on the process of converting tensors to matrices, and assumptions about the reader's familiarity with the underlying concepts may not be fully addressed.
Reedeegi said:How would you represent tensors as matrices? I've searched all over, and my book on GR (Wald) only has one example where he makes a matrice from a tensor, and I still don't understand the process.
George Jones said:A tensor is a muti-linear mappring. A rank 2 tensor can be turned into a matrix by letting it operate on a basis or dual basis. See,
https://www.physicsforums.com/showthread.php?p=874061&#post874061
but this might not have enough detail.
Reedeegi said:how often are rank-2 tensors used in GR, by chance?
George Jones said:Quite often; the metric tensor, the stress-energy tensor, the Einstein tensor, and the Ricci tensor all examples of rank 2 tensors.
Introducing a basis turns (the components of):
a rank 1 tensor into a column or row of values;
a rank 2 matrix into a square matrix of values;
a rank 3 tensor into a cube with values in each subcubes (think rubik's cube).