Undergrad Invariance of a tensor of order 2

[Thytanium]

TL;DR

In a tensor of order 1 (a vector) its magnitude, direction and sense must remain invariant given a change in coordinates; but in a tensor of order 2, I don't know what remains invariant.

Good morning friends of the Forum. For me it is difficult to geometrically imagine a tensor of order 2 and maybe that is why it is difficult for me to know, what remains invariant when making a change of coordinates of this tensor. The only thing I can think of it, is that since a tensor of order 2 is a tensor product of two vectors (Tensors of order 1), these two vectors remain invariant when changing coordinates and multiplying them tensorly. This I think is implicit in the definition of the order 2 tensor but I am not sure that these assertions I am making are correct. If you can clarify these doubts for me, I would appreciate it, friends of the Forum.

 

[Orodruin]

Good morning friends of the Forum.

Good evening!

For me it is difficult to geometrically imagine a tensor of order 2 and maybe that is why it is difficult for me to know, what remains invariant when making a change of coordinates of this tensor.

A tensor (including a vector) is a geometrical object. It is invariant by itself, what changes are the tensor components relative to whatever basis you choose to use.

The only thing I can think of it, is that since a tensor of order 2 is a tensor product of two vectors (Tensors of order 1), these two vectors remain invariant when changing coordinates and multiplying them tensorly.

More generally, a general tensor of order 2 is a linear combination of such products.

 

[Thytanium]

Thanks you friend Orodruin. Thanhs you very much.