I have very little experience and exposure to working with tensors. I know what a tensor is definitionally from a linear algebra course, and I know that rank 2 tensors transform in a particular way (this is the extent of my knowledge).

My question: Is there a way to determine if two tensors are the same even if their components are different?

This questions comes about because I am using mathematica to convert a 2-rank tensor from spherical coordinates to cartesian coordinates. I then converted the tensor back into spherical coordinates. The matrix I ended up with is different that the one I stated with. My guess is that this is happening because the rank 2 tensor transform multiplies one side by a matrix and the other by that matrix's transpose. Thus, performing this transformation and then performing the transform in the opposite way DOES NOT mean you're net multiplying the original tensor by identity matrices. So, perhaps you do not end up with what you started with.

Attached is a picture of my initial and end matrix (in spherical coordinates). All help would be greatly appreciated!

Edit: Added code used in case that is helpful.

[![enter image description here](https://i.stack.imgur.com/Aq1aq.png)](https://i.stack.imgur.com/Aq1aq.png)